1. ## Differential Equation question

The problems in the section of the book list the questions in this way:
dy/dx = 4y+e^4xsin(5x) and y(0)=1
First I get P(x)= -4 and then Q(x) = e^4x sin(5x)
then we put it in standard form and put the y on the left and the x on the right. Take the derivative first and then integrate. I understand the problems from #'s 18 on but the first 18 are set up like this:
y'-2xy= e^x^2 In the directions it states: primes denote derivatives with respect to x. The teacher explained the part in the book and with his example I understood and could do the later problems but I guess sometimes the hang up is what the question is asking. If someone has time to show me an example of the type y' -2xy= e^x^2 I would appreciate it. I assume we would do the same, get y on the left and x on the right and follow the same steps but not sure how to start. It seems that math is seeing patterns and if I could see one problem worked out I should be able to see the pattern and finish the rest of them.
Thank You,
Keith

2. Well, we're talking about linear ODEs. Do you know what the integrating factor is?

Originally Posted by keith
The problems in the section of the book list the questions in this way:
dy/dx = 4y+e^4xsin(5x) and y(0)=1
This is $\displaystyle y'-4y=e^{4x}\sin5x.$ (See my signature for LaTeX typesetting.)

Your integrating factor is $\displaystyle e^{-4x},$ so just multiply the entire equation by this term and go from there.

Originally Posted by keith
If someone has time to show me an example of the type y' -2xy= e^x^2 I would appreciate it.
The same idea: your integrating factor in this case is given by $\displaystyle \mu(x)=\exp\left\{\int-2x\,dx\right\}.$ Get it and do similar things with the first one.