Chapter 1 First-Order Differential Equations Alan H. Stein - - PowerPoint PPT Presentation

Chapter 1 – First-Order Differential Equations

Alan H. Stein University of Connecticut

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Separable Differential Equations

Any integral

• f (x) dx can be thought of as the general solution of

the differential equation dy dx = f (x).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Separable Differential Equations

Any integral

• f (x) dx can be thought of as the general solution of

the differential equation dy dx = f (x). There is a whole, slightly more general class of differential equations, called separable differential equations that can be solved almost as easily . . .

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Separable Differential Equations

Any integral

• f (x) dx can be thought of as the general solution of

the differential equation dy dx = f (x). There is a whole, slightly more general class of differential equations, called separable differential equations that can be solved almost as easily . . . with the understanding that the integration involved may not be at all easy, or even possible!

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Definition of a Separable D.E.

Definition (Separable Differential Equation)

A separable differential equation is one which can be written in the form f (x)dx dt = g(t).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Definition of a Separable D.E.

Definition (Separable Differential Equation)

A separable differential equation is one which can be written in the form f (x)dx dt = g(t). Note: The equation doesn’t have to be written in that form; it just has to be possible to rewrite it in that form.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Solutions of Separable Differential Equations

Given a separable differential equation f (x)dx dt = g(t), one may integrate both sides to get

• f (x)dx

dt dt =

• g(t) dt.

(*)

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Solutions of Separable Differential Equations

Given a separable differential equation f (x)dx dt = g(t), one may integrate both sides to get

• f (x)dx

dt dt =

• g(t) dt.

(*) Making the change of variable x = x(t), we dx = dx dt dt, so we may rewrite (*) as

• f (x) dx =
• g(t) dt.

(**)

Solutions of Separable Differential Equations

Given a separable differential equation f (x)dx dt = g(t), one may integrate both sides to get

• f (x)dx

dt dt =

• g(t) dt.

(*) Making the change of variable x = x(t), we dx = dx dt dt, so we may rewrite (*) as

• f (x) dx =
• g(t) dt.

(**) That may be viewed as the general solution.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Solutions of Separable Differential Equations

Given a separable differential equation f (x)dx dt = g(t), one may integrate both sides to get

• f (x)dx

dt dt =

• g(t) dt.

(*) Making the change of variable x = x(t), we dx = dx dt dt, so we may rewrite (*) as

• f (x) dx =
• g(t) dt.

(**) That may be viewed as the general solution. When we integrate on both sides of (**), we may get the solution x = φ(t) defined implicitly.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Slope Fields

Given a differential equation dy dt = f (t, y), at each selected point (t, y) in the ty-plane we draw a short line segment with slope f (t, y).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Slope Fields

Given a differential equation dy dt = f (t, y), at each selected point (t, y) in the ty-plane we draw a short line segment with slope f (t, y).

Special Cases

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Slope Fields

Given a differential equation dy dt = f (t, y), at each selected point (t, y) in the ty-plane we draw a short line segment with slope f (t, y).

Special Cases

dy dt = f (t) – the slope doesn’t change along a vertical line.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Slope Fields

Given a differential equation dy dt = f (t, y), at each selected point (t, y) in the ty-plane we draw a short line segment with slope f (t, y).

Special Cases

dy dt = f (t) – the slope doesn’t change along a vertical line. dy dt = f (y) – the slope doesn’t change along a horizontal line.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Slope Fields

Given a differential equation dy dt = f (t, y), at each selected point (t, y) in the ty-plane we draw a short line segment with slope f (t, y).

Special Cases

dy dt = f (t) – the slope doesn’t change along a vertical line. dy dt = f (y) – the slope doesn’t change along a horizontal line. Such differential equations are called autonomous.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Euler’s Method is a method of numerically approximating the solution to a differential equation of the form dy dt = f (t, y).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Euler’s Method is a method of numerically approximating the solution to a differential equation of the form dy dt = f (t, y). Euler’s Method is based on tangent approximations, using the fact that an equation of a line through a point (x0, y0) with slope m may be given by y = y0 + m(x − x0).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Euler’s Method is a method of numerically approximating the solution to a differential equation of the form dy dt = f (t, y). Euler’s Method is based on tangent approximations, using the fact that an equation of a line through a point (x0, y0) with slope m may be given by y = y0 + m(x − x0). This is a slight variation of the Point-Slope Formula y − y0 = m(x − x0),

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Euler’s Method is a method of numerically approximating the solution to a differential equation of the form dy dt = f (t, y). Euler’s Method is based on tangent approximations, using the fact that an equation of a line through a point (x0, y0) with slope m may be given by y = y0 + m(x − x0). This is a slight variation of the Point-Slope Formula y − y0 = m(x − x0), which itself comes directly from the definition m = y − y0 x − x0 of slope.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Assume we have a first order differential equation of the form dy dt = f (t, y) with intial conditions y(t0) = y0.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Assume we have a first order differential equation of the form dy dt = f (t, y) with intial conditions y(t0) = y0. Choose a step size h = ∆t.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Assume we have a first order differential equation of the form dy dt = f (t, y) with intial conditions y(t0) = y0. Choose a step size h = ∆t. Given a point (tk, yk), choose the next point (tk+1, yk+1) by letting tk+1 = tk + h = tk + ∆t and letting yk+1 = yk + f (tk, yk)h = yk + f (tk, yk)∆t.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

Assume we have a first order differential equation of the form dy dt = f (t, y) with intial conditions y(t0) = y0. Choose a step size h = ∆t. Given a point (tk, yk), choose the next point (tk+1, yk+1) by letting tk+1 = tk + h = tk + ∆t and letting yk+1 = yk + f (tk, yk)h = yk + f (tk, yk)∆t. Effectively, if (tk, yk) was on the graph of the solution, (tk+1, yk+1) would be on the line tangent to the graph, just like a tangent line approximation, otherwise known as a linear approximation or an approximation using differentials.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Euler’s Method

With Euler’s Method, xk is an approximation to x(ti).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Improvements on Euler’s Method

Euler’s Method can be improved, just as crude numerical integration methods like the Trapezoid Rule can be improved upon.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Improvements on Euler’s Method

Euler’s Method can be improved, just as crude numerical integration methods like the Trapezoid Rule can be improved upon. One way of improving is a general class called Predictor-Corrector methods.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Improvements on Euler’s Method

Euler’s Method can be improved, just as crude numerical integration methods like the Trapezoid Rule can be improved upon. One way of improving is a general class called Predictor-Corrector

• methods. With predictor-corrector methods, one uses a preliminary

estimate for (tk, yk) and then improves upon it.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Improvements on Euler’s Method

Euler’s Method can be improved, just as crude numerical integration methods like the Trapezoid Rule can be improved upon. One way of improving is a general class called Predictor-Corrector

• methods. With predictor-corrector methods, one uses a preliminary

estimate for (tk, yk) and then improves upon it. For example, one might let y∗

k+1 = yk + f (tk, yk)∆t and go back and use the

midpoint tk + tk+1 2 , yk + y∗

k+1

2

• , find the slope m = f (t, y) at

that point, and let yk+1 = yk + m∆t.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Existence and Uniqueness of Solutions

Theorem (Existence)

Suppose f (t, y) is continuous in a rectangle in the plane containing a point (t0, y0) in its interior. Then there exists some solution to the initial value problem dy dt = f (t, y), y(t0) = y0 in some open interval containing t0.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Existence and Uniqueness of Solutions

Theorem (Existence)

Suppose f (t, y) is continuous in a rectangle in the plane containing a point (t0, y0) in its interior. Then there exists some solution to the initial value problem dy dt = f (t, y), y(t0) = y0 in some open interval containing t0.

Theorem (Uniqueness Theorem)

Suppose we have a differential equations satisfying the hypotheses

• f the Existence Theorem and, additionally, ∂f

∂y is continuous in the same rectangle. Then the solution to the differential equation is unique.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Existence and Uniqueness of Solutions

Theorem (Existence)

Suppose f (t, y) is continuous in a rectangle in the plane containing a point (t0, y0) in its interior. Then there exists some solution to the initial value problem dy dt = f (t, y), y(t0) = y0 in some open interval containing t0.

Theorem (Uniqueness Theorem)

Suppose we have a differential equations satisfying the hypotheses

• f the Existence Theorem and, additionally, ∂f

∂y is continuous in the same rectangle. Then the solution to the differential equation is unique. This is very nice, since it essentially says the differential equation with initial conditions is enough to determine what happens.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

The Phase Line

For an autonomous differential equation, knowing the slope field along any vertical line suffices to know the slope field everywhere. We can draw just one vertical line, called the phase line, containing all that information.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

The Phase Line

For an autonomous differential equation, knowing the slope field along any vertical line suffices to know the slope field everywhere. We can draw just one vertical line, called the phase line, containing all that information. We also only need to know whether the slope is positive or negative to get a good, qualitative idea of the nature

• f solutions.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

The Phase Line

For an autonomous differential equation, knowing the slope field along any vertical line suffices to know the slope field everywhere. We can draw just one vertical line, called the phase line, containing all that information. We also only need to know whether the slope is positive or negative to get a good, qualitative idea of the nature

• f solutions.

We draw phase lines which show the equilibrium points along with the sign of the derivative in between equilibrium points.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Drawing Phase Lines

Given an autonomous differential equation dy dt = f (y), we draw the phase line using the following steps:

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Drawing Phase Lines

Given an autonomous differential equation dy dt = f (y), we draw the phase line using the following steps:

◮ Draw the y-axis.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Drawing Phase Lines

Given an autonomous differential equation dy dt = f (y), we draw the phase line using the following steps:

◮ Draw the y-axis. ◮ Find the equilibrium points (where f (y) = 0) and mark them

• n the axis.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Drawing Phase Lines

Given an autonomous differential equation dy dt = f (y), we draw the phase line using the following steps:

◮ Draw the y-axis. ◮ Find the equilibrium points (where f (y) = 0) and mark them

• n the axis.

◮ In each interval determined by the equilibrium points,

determine whether f (y) is positive or negative.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Drawing Phase Lines

Given an autonomous differential equation dy dt = f (y), we draw the phase line using the following steps:

◮ Draw the y-axis. ◮ Find the equilibrium points (where f (y) = 0) and mark them

• n the axis.

◮ In each interval determined by the equilibrium points,

determine whether f (y) is positive or negative.

◮ In each interval where f (y) is positive, draw an arrow pointing

up.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Drawing Phase Lines

Given an autonomous differential equation dy dt = f (y), we draw the phase line using the following steps:

◮ Draw the y-axis. ◮ Find the equilibrium points (where f (y) = 0) and mark them

• n the axis.

◮ In each interval determined by the equilibrium points,

determine whether f (y) is positive or negative.

◮ In each interval where f (y) is positive, draw an arrow pointing

up.

◮ In each interval where f (y) is negative, draw an arrow pointing

down.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Classification of Equilibrium Points

Equilibrium points are classified as sources, sinks or nodes.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Classification of Equilibrium Points

Equilibrium points are classified as sources, sinks or nodes.

◮ Source – An equilibrium point y0 is a source if any solution

sufficiently close to y = y0 is asymptotic to y = y0 as t → −∞.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Classification of Equilibrium Points

Equilibrium points are classified as sources, sinks or nodes.

◮ Source – An equilibrium point y0 is a source if any solution

sufficiently close to y = y0 is asymptotic to y = y0 as t → −∞.

◮ Sink – An equilibrium point y0 is a sink if any solution

sufficiently close to y = y0 is asymptotic to y = y0 as t → ∞.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Classification of Equilibrium Points

Equilibrium points are classified as sources, sinks or nodes.

◮ Source – An equilibrium point y0 is a source if any solution

sufficiently close to y = y0 is asymptotic to y = y0 as t → −∞.

◮ Sink – An equilibrium point y0 is a sink if any solution

sufficiently close to y = y0 is asymptotic to y = y0 as t → ∞.

◮ Node – An equilibrium point which is neither a source nor a

sink is a node.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink;

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink; ◮ If f ′(y0) > 0, then y0 is a source;

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink; ◮ If f ′(y0) > 0, then y0 is a source; ◮ If f ′(y0) = 0, y0 can be a sink, source or node.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink; ◮ If f ′(y0) > 0, then y0 is a source; ◮ If f ′(y0) = 0, y0 can be a sink, source or node.

To see why: If y0 is an equilibrium point, then f (y0) = 0. If f ′(y0) < 0, then f must be decreasing near y0.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink; ◮ If f ′(y0) > 0, then y0 is a source; ◮ If f ′(y0) = 0, y0 can be a sink, source or node.

To see why: If y0 is an equilibrium point, then f (y0) = 0. If f ′(y0) < 0, then f must be decreasing near y0. It follows that, nearby, dy dt = f (y) > 0 for y < y0, while dy dt = f (y) < 0 for y > y0,

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink; ◮ If f ′(y0) > 0, then y0 is a source; ◮ If f ′(y0) = 0, y0 can be a sink, source or node.

To see why: If y0 is an equilibrium point, then f (y0) = 0. If f ′(y0) < 0, then f must be decreasing near y0. It follows that, nearby, dy dt = f (y) > 0 for y < y0, while dy dt = f (y) < 0 for y > y0, making y0 a sink.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linearization Theorem

Theorem (Linearization Theorem)

Let y0 be an equilibrium point of the differential equation dy dt = f (y).

◮ If f ′(y0) < 0, then y0 is a sink; ◮ If f ′(y0) > 0, then y0 is a source; ◮ If f ′(y0) = 0, y0 can be a sink, source or node.

To see why: If y0 is an equilibrium point, then f (y0) = 0. If f ′(y0) < 0, then f must be decreasing near y0. It follows that, nearby, dy dt = f (y) > 0 for y < y0, while dy dt = f (y) < 0 for y > y0, making y0 a sink. Similar logic holds if f ′(y0) > 0.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Equations With Parameters

We may have a family of differential equations which include a paramter distinguishing them.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Equations With Parameters

We may have a family of differential equations which include a paramter distinguishing them. Examples:

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Equations With Parameters

We may have a family of differential equations which include a paramter distinguishing them. Examples:

◮ dx

dt = kx

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Equations With Parameters

We may have a family of differential equations which include a paramter distinguishing them. Examples:

◮ dx

dt = kx

◮ dy

dt = y2 − 2y + µ

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Equations With Parameters

We may have a family of differential equations which include a paramter distinguishing them. Examples:

◮ dx

dt = kx

◮ dy

dt = y2 − 2y + µ In general, we may refer to a the differential equation dy dt = fµ(y) as a one-parameter family of differential equations.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Bifurcations

Usually, a small change in the parameter will not result in a significant change in the nature of the solutions. Usually, changing the parameter slightly will merely result in a small change in the equilibrium points. One way of thinking about this is to look at the phase line.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Bifurcations

Usually, a small change in the parameter will not result in a significant change in the nature of the solutions. Usually, changing the parameter slightly will merely result in a small change in the equilibrium points. One way of thinking about this is to look at the phase line. Sometimes, a small change in the parameter results in a drastic change in the phase line and the nature of some of the solutions. If the nature of the phase line is different for µ < µ0 and µ > µ0, then µ0 is called a bifurcation value.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Bifurcation Diagram

One visual help is a Bifurcation Diagram. We get a bifurcation diagram by using the µy-plane and drawing phase lines for different values of µ.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Sources and Sinks

We can observe if dy dt = fµ0(y) has an equilibrium point at y = y0 and f ′

µ0(y0) < 0, then the equilibrium point is a sink. If for some

µ1 very close to µ0, the equation dy dt = fµ1(y) has a nearby equilibrium point, f ′

µ1(y) would have to be negative at that

equilibrium point as well, so that would also be a sink, so that no bifurcation could have occurred.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Sources and Sinks

We can observe if dy dt = fµ0(y) has an equilibrium point at y = y0 and f ′

µ0(y0) < 0, then the equilibrium point is a sink. If for some

µ1 very close to µ0, the equation dy dt = fµ1(y) has a nearby equilibrium point, f ′

µ1(y) would have to be negative at that

equilibrium point as well, so that would also be a sink, so that no bifurcation could have occurred. We can similarly observe no bifurcation could occur near an equilibrium point y0 for which f ′

µ0(y) > 0.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Sources and Sinks

We can observe if dy dt = fµ0(y) has an equilibrium point at y = y0 and f ′

µ0(y0) < 0, then the equilibrium point is a sink. If for some

µ1 very close to µ0, the equation dy dt = fµ1(y) has a nearby equilibrium point, f ′

µ1(y) would have to be negative at that

equilibrium point as well, so that would also be a sink, so that no bifurcation could have occurred. We can similarly observe no bifurcation could occur near an equilibrium point y0 for which f ′

µ0(y) > 0.

It follows that, for a bifurcation value to exist, we need an equilibrium point y = y0 and µ = µ0 where fµ0(y0) = 0 and f ′

µ0(y0) = 0

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linear Differential Equations

Definition (First Order Linear Differential Equation)

A first order linear differential equation is a differential equation which can be written in the form dy dt + α(t)y = r(t).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Linear Differential Equations

Definition (First Order Linear Differential Equation)

A first order linear differential equation is a differential equation which can be written in the form dy dt + α(t)y = r(t). First Order Linear Differential Equations may be solved through the use of integrating factors.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Integrating Factors

The idea behind an integrating factor is to find a function µ = µ(t), called an integrating factor, so that if we multiply both sides of the differential equation dy dt + α(t)y = r(t) by µ(t),

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Integrating Factors

The idea behind an integrating factor is to find a function µ = µ(t), called an integrating factor, so that if we multiply both sides of the differential equation dy dt + α(t)y = r(t) by µ(t), to get the differential equation µ(t)dy dt + µ(t)α(t)y = µ(t)r(t), then the left-hand side is the derivative of d dt (µy) = d dt (µ(t)y).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Integrating Factors

The idea behind an integrating factor is to find a function µ = µ(t), called an integrating factor, so that if we multiply both sides of the differential equation dy dt + α(t)y = r(t) by µ(t), to get the differential equation µ(t)dy dt + µ(t)α(t)y = µ(t)r(t), then the left-hand side is the derivative of d dt (µy) = d dt (µ(t)y). If we can find such an integrating factor, we can then rewrite the differential equation in the form d dt (µ(t)y) = µ(t)r(t)

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Integrating Factors

The idea behind an integrating factor is to find a function µ = µ(t), called an integrating factor, so that if we multiply both sides of the differential equation dy dt + α(t)y = r(t) by µ(t), to get the differential equation µ(t)dy dt + µ(t)α(t)y = µ(t)r(t), then the left-hand side is the derivative of d dt (µy) = d dt (µ(t)y). If we can find such an integrating factor, we can then rewrite the differential equation in the form d dt (µ(t)y) = µ(t)r(t) and this can be solved by integrating.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to find a function µ = µ(t) such that d dt (µy) = µdy dt + µαy.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to find a function µ = µ(t) such that d dt (µy) = µdy dt + µαy. Since d dt (µy) = µdy dt + dµ dt y,

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to find a function µ = µ(t) such that d dt (µy) = µdy dt + µαy. Since d dt (µy) = µdy dt + dµ dt y, we need to solve µdy dt + µαy = µdy dt + dµ dt y.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to find a function µ = µ(t) such that d dt (µy) = µdy dt + µαy. Since d dt (µy) = µdy dt + dµ dt y, we need to solve µdy dt + µαy = µdy dt + dµ dt y. Simplifying, we may equate µαy = dµ dt y,

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to find a function µ = µ(t) such that d dt (µy) = µdy dt + µαy. Since d dt (µy) = µdy dt + dµ dt y, we need to solve µdy dt + µαy = µdy dt + dµ dt y. Simplifying, we may equate µαy = dµ dt y, or simply dµ dt = µα.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to find a function µ = µ(t) such that d dt (µy) = µdy dt + µαy. Since d dt (µy) = µdy dt + dµ dt y, we need to solve µdy dt + µαy = µdy dt + dµ dt y. Simplifying, we may equate µαy = dµ dt y, or simply dµ dt = µα. This is a separable differential equation which we can solve relatively routinely.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to solve dµ dt = µα. Remember, both µ and α are functions of t.

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to solve dµ dt = µα. Remember, both µ and α are functions of t. We may solve as follows:

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to solve dµ dt = µα. Remember, both µ and α are functions of t. We may solve as follows: 1 µ dµ dt = α

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to solve dµ dt = µα. Remember, both µ and α are functions of t. We may solve as follows: 1 µ dµ dt = α 1 µ dµ =

• α(t) dt

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to solve dµ dt = µα. Remember, both µ and α are functions of t. We may solve as follows: 1 µ dµ dt = α 1 µ dµ =

• α(t) dt

ln |µ| =

• α(t) dt

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Finding an Integrating Factor

We need to solve dµ dt = µα. Remember, both µ and α are functions of t. We may solve as follows: 1 µ dµ dt = α 1 µ dµ =

• α(t) dt

ln |µ| =

• α(t) dt

µ = exp(

• α(t) dt)

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Using the Integrating Factor

We now have the differential equation dy dt + α(t)y = r(t) and the integrating factor µ = exp(

• α(t) dt).

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Using the Integrating Factor

We now have the differential equation dy dt + α(t)y = r(t) and the integrating factor µ = exp(

• α(t) dt). We may solve as follows:

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Using the Integrating Factor

We now have the differential equation dy dt + α(t)y = r(t) and the integrating factor µ = exp(

• α(t) dt). We may solve as follows:

µdy dt + µα(t)y = µr(t)

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Using the Integrating Factor

We now have the differential equation dy dt + α(t)y = r(t) and the integrating factor µ = exp(

• α(t) dt). We may solve as follows:

µdy dt + µα(t)y = µr(t) d dt (µy) = µr(t)

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Using the Integrating Factor

We now have the differential equation dy dt + α(t)y = r(t) and the integrating factor µ = exp(

• α(t) dt). We may solve as follows:

µdy dt + µα(t)y = µr(t) d dt (µy) = µr(t) µy =

• µ(t)r(t) dt + c

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations

Using the Integrating Factor

We now have the differential equation dy dt + α(t)y = r(t) and the integrating factor µ = exp(

• α(t) dt). We may solve as follows:

µdy dt + µα(t)y = µr(t) d dt (µy) = µr(t) µy =

• µ(t)r(t) dt + c

y = 1 µ(

• µ(t)r(t) dt + c)

Alan H. SteinUniversity of Connecticut Chapter 1 – First-Order Differential Equations