# At a Glance - Solving Differential Equations

Checking to see if a given function satisfies a given differential equation isn't too horrible of a task (at least for the functions we encounter in Calculus).

The task of solving differential equations from scratch is a bit different.

Differential equations fall into several very, very broad categories:

• ones you can solve right now by thinking backwards.

• ones you'll be able to solve by the end of the calculus class.

• ones you'll be able to solve if you take other courses about differential equations, and

• ones that people at places like MIT are still working on.

We need to ignore most of these for now and concentrate on the ones we can solve by thinking backwards.

#### Example 1

 Find a solution to the differential equationy' = 2x.

#### Example 2

 Find all solutions to the differential equationy' = 2x.

#### Example 3

 Find a solution to the differential equation

#### Example 4

 Find a solution to the d.e. y" = 5.

#### Example 5

 Find all solutions to the d.e. y" = 5.

#### Exercise 1

For the differential equation, find (a) one solution and then (b) all solutions.

y ' = 4x3

#### Exercise 2

For the differential equation, find (a) one solution and then (b) all solutions.

y' = x5

#### Exercise 3

For the differential equation, find (a) one solution and then (b) all solutions.

#### Exercise 4

For the differential equation, find (a) one solution and then (b) all solutions.

xy' = 4x

#### Exercise 5

For the differential equation, find (a) one solution and then (b) all solutions.

y" = -x-2