Directly Integrable Equations

When a derivative can be written as a function of the independent variable only for example

y'=f(x)

y''=f(x)

y'''(x)=f(x)

etc.

We call that a directly integrable equation.

Example 1:

Find the general solution for the following directly integrable equation

That means

 c is an arbitary solution

And  if we have the initial condition

y(1)=3

So we will need to get the y' by itself

so,

 , where c is an arbitary constant

And this is our general solution. Now we plug in the initial condition when .

That means

Thus, the particular solution is