Please help with differential equations

4.15

is a linear ODE; multiply both sides by the integrating factor :

Now the left side can be condensed as the derivative of a product:

Integrate both sides and solve for to get

Given that , we find

and so the particular solution is

5.15

You may be tempted to write this as an ODE in , but the ODE in is much easier to solve, since it's linear. Solve for and rearrange the terms:

Multiply both sides by the integrating factor , then solve for :

Given that when , we have

so the particular solution is

or, by solving for ,

6.15

Dividing through both sides by lets us write the equation in Bernoulli form:

Substitute , so that . Then we get an ODE that is linear in :

Multiply both sides by the integrating factor :

Integrate both sides and solve for , then solve for :

Given that , we find

so the particular solution is

-6 is the correct option from the above given options