find the gen. solution of dy/dx + ylogy = ye^x
using the substituion u = logy
Let's do the substitution :
Substitute and in the equation :
Simplify by and the equation becomes
This is a first order ODE hence you've to solve the homogeneous equation first : . It'll give a solution . Then, you need to find a particular solution of . As the second member is , you may try . (find by replacing by its expression in ) The solutions of will be given by the sum and then you can use to get the solutions in terms of .
The first thing to do is to solve the homogeneous equation : the solutions are
Then, we have to find a particular solution of . Using my own advice, I try with . Let's replace this in the equation and solve for :
Simplifying both sides by , we get whence
The solutions of the ODE in terms of are given by
Using , we get the solutions of :
and replied too.