Differential Equations

Separable Differential Equations

A separable differential equation can be expressed as the product of a

function of x and a function of y.

dyg x h y

dx

Example:

22dy

xydx

Multiply both sides by dx and divide both sides by y2 to separate the variables. (Assume y2 is never zero.)

22

dyx dx

y

2 2 y dy x dx

0h y

A separable differential equation can be expressed as the product of a

function of x and a function of y.

dyg x h y

dx

Example:

22dy

xydx

22

dyx dx

y

2 2 y dy x dx

2 2 y dy x dx

21x C

y

2

1y

x C

0h y

Separable Differential Equations

Example:

222 1 xdyx y e

dx

2

2

12

1xdy x e dx

y

Separable differential equation

2

2

12

1xdy x e dx

y

2u x

2

1

1udy e du

y

Solve the differential with initial condition

Example (cont.):

222 1 xdyx y e

dx

21tan xy e C We now have y as an implicit function of x.

We can find y as an explicit function of x by taking the tangent of both sides.

21tan tan tan xy e C

2

tan xy e C

A harder problem

Careful when separating, wecan only separateby multiplyingor dividing!!!

A Trickier problem!