a) Find the velocity of the particle at t=8 seconds.

a) Find the position of the particle at t=4 seconds.

WARMUP

Separable Differential Equations

Review: Find .

Example: Solve the differential equation.

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

Combined constants of integration

22dy

xydx

Example 9:

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 9:

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.