General solution of differential equation

[captainjack2000]

1. The question asks me to show that e^x is a solution of xy'' - (2x+1)y' + (x+1)y=0 and find the general solution.



2. I managed to simplify the equation to u''xe^(x) - u'e^(x) = 0 by letting y=ue^(x) and finding the differentials and substituting them in.
I've then let z=u dz/du=u' and d^2z/du^2 = u''
so I get xe^(x)(d^2z/du^2) - e^(x)dz/du = 0
How would I solve this?

 

[kring_c14]

use integration by parts

 

[captainjack2000]

Sorry but I am still confused;

if xe^(x)(d^2z/du^2) - e^(x)dz/du = 0
can I simplify this to

x(d^2z/du^2) = dz/du

xdz = du
xz = u +c but z = u?
xu = u +c
How does this help find the general solution?

 

[HallsofIvy]

1. The question asks me to show that e^x is a solution of xy'' - (2x+1)y' + (x+1)y=0 and find the general solution.



2. I managed to simplify the equation to u''xe^(x) - u'e^(x) = 0 by letting y=ue^(x) and finding the differentials and substituting them in.
I've then let z=u dz/du=u' and d^2z/du^2 = u''
so I get xe^(x)(d^2z/du^2) - e^(x)dz/du = 0
How would I solve this?

You don't solve that! It has two independent variables, u and x. Surely, you don't mean "let z=u dz/du=u' and d^2z/du^2 = u''. If you let z= u, then dz/du= 1. Perhaps you meant "let z= u, dz/dx= u'". But in that case you've gained nothing- you've just renamed u. Much better is "let z= du/dx, dz/dx= d²u/dx²". Then your equation becomes the first order equation xe^(x) z'- e^(x) z= 0. I would be inclined to first divide the entire equation by e^(x)!

 

[doroulla]

Hi can someone please help me with: general solution of dy/dx - y = x + 2x^2

i know how to find general solutions but only when i can separate the y and x in 2 sides and multiply with dx and dy. someone please help me. i have looked in many books

 

[HallsofIvy]

It is NOT good idea to add on to someone elses's thread. People who have already responded to the thread may not even look at your post. Use the "new thread" button on the main menu. In this case, I have already responded on the thread you did start about 4 minutes later!