6.4 day 1 Separable Differential Equations

Jefferson Memorial, Washington DCGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007

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

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

22

dyx dx

y

2 2 y dy x dx

2 2 y dy x dx 1 2

1 2y C x C

21x C

y

2

1y

x C

2

1y

x C

0h y

Combined constants of integration

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 du x dx

2

1

1udy e du

y

1

1 2tan uy C e C 21

1 2tan xy C e C 21tan xy e C Combined constants of integration

Example:

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

Notice that we can not factor out the constant C, because the distributive property does not work with tangent.