Separable Differential Equations Bernd Schr oder logo1 Bernd Schr - - PowerPoint PPT Presentation

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Separable Differential Equations

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

What are Separable Differential Equations?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

What are Separable Differential Equations?

• 1. A separable differential equation is of the form

y′ = f(x)g(y).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

What are Separable Differential Equations?

• 1. A separable differential equation is of the form

y′ = f(x)g(y).

• 2. That is, a differential equation is separable if the terms that

are not equal to y′ can be factored into a factor that only depends on x and another factor that only depends on y.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

What are Separable Differential Equations?

• 1. A separable differential equation is of the form

y′ = f(x)g(y).

• 2. That is, a differential equation is separable if the terms that

are not equal to y′ can be factored into a factor that only depends on x and another factor that only depends on y.

• 3. The solution method for separable differential equations

looks like regular algebra with the added caveat that we use integrals to undo the differentials dx and dy from y′ = dy dx.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey dy ey = x dx

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx
• e−y dy

=

• x dx

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx
• e−y dy

=

• x dx

−e−y = 1 2x2 +c

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx
• e−y dy

=

• x dx

−e−y = 1 2x2 +c e−y = −c− 1 2x2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx
• e−y dy

=

• x dx

−e−y = 1 2x2 +c e−y = C − 1 2x2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx
• e−y dy

=

• x dx

−e−y = 1 2x2 +c e−y = C − 1 2x2 −y = ln

• C − 1

2x2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

dy dx = xey

dy

ey =

• x dx
• e−y dy

=

• x dx

−e−y = 1 2x2 +c e−y = C − 1 2x2 −y = ln

• C − 1

2x2

• y

= −ln

• C − 1

2x2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

The general solution is a family of functions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

The general solution is a family of functions. y = −ln

• C − 1

2x2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

The general solution is a family of functions. y = −ln

• C − 1

2x2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

The general solution is a family of functions. y = −ln

• C − 1

2x2

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

◮ The general solution of the differential equation y′ = xey is

y = −ln

• C − 1

2x2

• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

◮ The general solution of the differential equation y′ = xey is

y = −ln

• C − 1

2x2

• .

◮ But how do we know that the above is right?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

◮ The general solution of the differential equation y′ = xey is

y = −ln

• C − 1

2x2

• .

◮ But how do we know that the above is right?

Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

◮ The general solution of the differential equation y′ = xey is

y = −ln

• C − 1

2x2

• .

◮ But how do we know that the above is right?

Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution.

◮ But how do we know that we did not make any mistakes?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

◮ The general solution of the differential equation y′ = xey is

y = −ln

• C − 1

2x2

• .

◮ But how do we know that the above is right?

Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution.

◮ But how do we know that we did not make any mistakes?

We don’t.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Solve the differential equation y′ = xey.

◮ The general solution of the differential equation y′ = xey is

y = −ln

• C − 1

2x2

• .

◮ But how do we know that the above is right?

Although the method looks a bit fishy (you’re not supposed to separate differentials) there are theorems which guarantee that it gives the general solution.

◮ But how do we know that we did not make any mistakes?

We don’t.

◮ So we should always check the result.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey d dx

• −ln
• C − 1

2x2

• ?

= xe−ln(C− 1

2x2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey d dx

• −ln
• C − 1

2x2

• ?

= xe−ln(C− 1

2x2)

− 1 C − 1

2x2(−x) ?

=

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey d dx

• −ln
• C − 1

2x2

• ?

= xe−ln(C− 1

2x2)

− 1 C − 1

2x2(−x) ?

= x 1 eln(C− 1

2x2)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey d dx

• −ln
• C − 1

2x2

• ?

= xe−ln(C− 1

2x2)

− 1 C − 1

2x2(−x) ?

= x 1 eln(C− 1

2x2)

x 1 C − 1

2x2 ?

=

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey d dx

• −ln
• C − 1

2x2

• ?

= xe−ln(C− 1

2x2)

− 1 C − 1

2x2(−x) ?

= x 1 eln(C− 1

2x2)

x 1 C − 1

2x2 ?

= x 1 C − 1

2x2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations

logo1 Definition An Example Double Check

Does y = −ln

• C − 1

2x2

• Really Solve the

Differential Equation y′ = xey?

y′

?

= xey d dx

• −ln
• C − 1

2x2

• ?

= xe−ln(C− 1

2x2)

− 1 C − 1

2x2(−x) ?

= x 1 eln(C− 1

2x2)

x 1 C − 1

2x2

= x 1 C − 1

2x2

√

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separable Differential Equations