It's separable: so I'd just let and end up with which doesn't look bad to integrate via parts.
I think the following DE is of 1st order, linear and homogeneous, correct me if I'm wrong.
I don't know which method to use to solve it.
Using the integrating factor method, I have to calculate , which is a problem to me.
I'm having problems with the IBP. What do you choose as f and g'?
If I left and , then I'll have to calculate for which I've no clue about how to calculate it. I tried another IBP but with no success.
If I let and , I'll have to calculate , which is worse than the original integral.
So I'd opt for the first option, but I've some problems solving . Am I in the right direction? Am I missing something?
Ok, I made a mistake. Sorry. Not easy to integrate that. The reason I say that is I just used DSolve in Mathematica and it returns FrenselC functions which is just another name for the integral encountered in this problem but I personally think is ok in terms of an "exact" solution. So then I'd leave it as:
or if you wish:
Ok, let me try to explain without getting into more trouble: That FresnelC thing is a Mathematica construct for the Fresnel integral and that's why the particular example above has the in it. Really, we started with:
which I get by parts:
and if and we use the standard definition of a Fresnel integral: then I could write the solution as: