Warm Up

Solve the initial value problem

7.4 A – Separable Differential Equations•Use separation and initial values to solve differential

equations.

Definition

•A differential equation of the form is called separable.

•We separate the variables by writing it in the form

Steps

1. Group all y’s on one side of the equation with dy.2. Group all the x’s on the other side with dx. 3. Integrate both sides with respect to their variable.4. Note* Only one constant (C) is needed. 5. Apply the initial condition and solve for C.6. Plug C back in, and solve for y. 7. Your goal is to get an equation in the form y=h(x)8. Find the domain.

Example

•Solve for y if when x=1 and y=1.

S o l.

•This is separable because it can be written as where

•Now, integrate both sides.

Sol.

•Apply the initial condition to find C.

Sol.

•Plug C in and solve for y.

To find the domain:

•Ask “Where does this function not exist?”

• Set denominator =0.

•Domain:

You try!

when y(1)=0

Example 2

•

(no initial value, so there will be a C in the answer.

Sol.

• Show work on board (from notes)

Homework

• Only 1-10.