How can I effectively solve two challenging differential equations problems?

[FaradayLaws]

Homework Statement

I have two problems from my differential equations assignment that I'm having difficulty with. I would appreciate some guidance!

Homework Equations

http://img10.imageshack.us/img10/3397/questionsk.th.jpg

The Attempt at a Solution

for no.10 I used reduction of order with the assumption that one solution is y1=e^x
I got y2=xe^x

my question is for this question do I solve the unhomogenous equation by variation of parameters to solve for the particular solution And from there use it for the General Solution ? ( yg= yh+yp.

For no.7
I multiplied by x^p y^q and found the partial derivatives My and Nx.
Inorder for it to be exact, I equated the partial derivatives and found my values for p and q. They came out to being fractions and my final solution is extremely messy with fractions. Is this is correct approach?

Thanks.

 

[HallsofIvy]

Homework Statement

I have two problems from my differential equations assignment that I'm having difficulty with. I would appreciate some guidance!

Homework Equations

http://img10.imageshack.us/img10/3397/questionsk.th.jpg

The Attempt at a Solution

for no.10 I used reduction of order with the assumption that one solution is y1=e^x
I got y2=xe^x

my question is for this question do I solve the unhomogenous equation by variation of parameters to solve for the particular solution And from there use it for the General Solution ? ( yg= yh+yp.

Yes.

for no.7
I multiplied by x^p y^q and found the partial derivatives My and Nx.
Inorder for it to be exact, I equated the partial derivatives and found my values for p and q. They came out to being fractions and my final solution is extremely messy with fractions. Is this is correct approach?

Did you get 3q= 2(p+1) and 20(q-1)= 12(p+ 3)? That's what I got. Yes, those give "messy" fractions.

Thanks.[/QUOTE]

 

[Architeuthis Dux]

I would suggest first reviewing the concepts and techniques for solving differential equations, such as reduction of order and variation of parameters. These are important tools for solving differential equations and understanding the solutions. Additionally, it would be helpful to review the steps for solving exact equations, as mentioned in the Attempt at a Solution for problem no. 7. It is important to carefully check your work and make sure your final solution makes sense mathematically. If you are still having difficulty, it may be helpful to seek assistance from a tutor or your professor. Practice and repetition are key in mastering differential equations, so don't get discouraged and keep working at it. Good luck with your assignment!