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 ( x 0 , y 0 ) with slope m may be given by y = y 0 + m ( x − x 0 ). 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 ( x 0 , y 0 ) with slope m may be given by y = y 0 + m ( x − x 0 ). This is a slight variation of the Point-Slope Formula y − y 0 = m ( x − x 0 ), 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 ( x 0 , y 0 ) with slope m may be given by y = y 0 + m ( x − x 0 ). This is a slight variation of the Point-Slope Formula y − y 0 = m ( x − x 0 ), which itself comes directly from the definition m = y − y 0 x − x 0 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 ( t 0 ) = y 0 . 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 ( t 0 ) = y 0 . 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 ( t 0 ) = y 0 . Choose a step size h = ∆ t . Given a point ( t k , y k ), choose the next point ( t k +1 , y k +1 ) by letting t k +1 = t k + h = t k + ∆ t and letting y k +1 = y k + f ( t k , y k ) h = y k + f ( t k , y k )∆ 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 ( t 0 ) = y 0 . Choose a step size h = ∆ t . Given a point ( t k , y k ), choose the next point ( t k +1 , y k +1 ) by letting t k +1 = t k + h = t k + ∆ t and letting y k +1 = y k + f ( t k , y k ) h = y k + f ( t k , y k )∆ t . Effectively, if ( t k , y k ) was on the graph of the solution, ( t k +1 , y k +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, x k is an approximation to x ( t i ). 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