find the gen. solution of dy/dx + ylogy = ye^x

using the substituion u = logy

thankyou!

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- May 11th 2008, 01:11 AMmathshelpbookcalculus - homogenous de
find the gen. solution of dy/dx + ylogy = ye^x

using the substituion u = logy

thankyou! - May 11th 2008, 02:25 AMflyingsquirrel
Hi

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 . - May 11th 2008, 02:51 AMmathshelpbook
thanks for your help! ok so i understand up to du/dx + u = e^x, but then i'm confused as to how to continue solving?

- May 11th 2008, 03:51 AMflyingsquirrel
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 : - May 11th 2008, 07:04 AMmr fantastic
Also asked by a different member here: http://www.mathhelpforum.com/math-he...tion-help.html

and replied too.