Separable Differential Equations - USM...logo1 Definition An Example Double Check What are Separable Differential Equations? 1. A separable differential equation is of the form y0

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Definition An Example Double Check

Separable Differential Equations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

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Definition An Example Double Check

What are Separable Differential Equations?

1. A separable differential equation is of the formy′ = f (x)g(y).

2. That is, a differential equation is separable if the terms thatare not equal to y′ can be factored into a factor that onlydepends on x and another factor that only depends on y.

3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use

integrals to undo the differentials dx and dy from y′ =dydx

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

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

3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use

integrals to undo the differentials dx and dy from y′ =dydx

.

Bernd Schroder 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 onlydepends on x and another factor that only depends on y.

3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use

integrals to undo the differentials dx and dy from y′ =dydx

.

Bernd Schroder 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 onlydepends on x and another factor that only depends on y.

3. The solution method for separable differential equationslooks like regular algebra with the added caveat that we use

integrals to undo the differentials dx and dy from y′ =dydx

.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

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Definition An Example Double Check

Solve the differential equation y′ = xey.

dydx

= xey

dyey = x dx

∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey

dyey = x dx

∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey

dyey = x dx

∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx

∫e−y dy =

∫x dx

−e−y =12

x2 + c

e−y = −c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = −c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = −c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = −c− 12

x2

−y = ln(

c− 12

x2)

y = − ln(

c− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = C− 12

x2

−y = ln(

C− 12

x2)

y = − ln(

C− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = C− 12

x2

−y = ln(

C− 12

x2)

y = − ln(

C− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.dydx

= xey∫ dyey =

∫x dx∫

e−y dy =∫

x dx

−e−y =12

x2 + c

e−y = C− 12

x2

−y = ln(

C− 12

x2)

y = − ln(

C− 12

x2)

Bernd Schroder 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− 12

x2)

Bernd Schroder 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− 12

x2)

Bernd Schroder 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− 12

x2)

Bernd Schroder 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− 12

x2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.

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

y =− ln(

C− 12

x2)

.

I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.

I But how do we know that we did not make any mistakes?We don’t.

I So we should always check the result.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.

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

y =− ln(

C− 12

x2)

.

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

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

I But how do we know that we did not make any mistakes?We don’t.

I So we should always check the result.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.

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

y =− ln(

C− 12

x2)

.

I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.

I But how do we know that we did not make any mistakes?We don’t.

I So we should always check the result.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.

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

y =− ln(

C− 12

x2)

.

I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.

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

We don’t.I So we should always check the result.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.

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

y =− ln(

C− 12

x2)

.

I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.

I But how do we know that we did not make any mistakes?We don’t.

I So we should always check the result.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Solve the differential equation y′ = xey.

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

y =− ln(

C− 12

x2)

.

I But how do we know that the above is right?Although the method looks a bit fishy (you’re not supposedto separate differentials) there are theorems whichguarantee that it gives the general solution.

I But how do we know that we did not make any mistakes?We don’t.

I So we should always check the result.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?=

x1

eln(C− 12 x2)

x1

C− 12x2

?= x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?=

x1

eln(C− 12 x2)

x1

C− 12x2

?= x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?=

x1

eln(C− 12 x2)

x1

C− 12x2

?= x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?=

x1

eln(C− 12 x2)

x1

C− 12x2

?= x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?= x

1

eln(C− 12 x2)

x1

C− 12x2

?=

x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?= x

1

eln(C− 12 x2)

x1

C− 12x2

?=

x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?= x

1

eln(C− 12 x2)

x1

C− 12x2

?= x1

C− 12x2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations

logo1

Definition An Example Double Check

Does y =− ln(

C− 12

x2)

Really Solve the

Differential Equation y′ = xey?

y′ ?= xey

ddx

[− ln

(C− 1

2x2

)]?= xe− ln(C− 1

2 x2)

− 1C− 1

2x2(−x) ?= x

1

eln(C− 12 x2)

x1

C− 12x2

= x1

C− 12x2

√

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separable Differential Equations