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Need help with differential equation.

by Ammar Kurd
Tags: diff equation, problem, verify that
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Ammar Kurd
#1
May21-12, 03:56 PM
P: 3
Hello everyone, this is my first post in "Physics Forums"...

I need help with this problem:

Verify that y = (x^2/2) + ((x/2)*√(x^2 + 1)) + ln(√(x+√(x^2 +1))) is a solution of the

equation 2y = x*y' + ln(y').....(1)

I differentiated (y) with respect to (x), and substituted (y' and y) in equation (1), but that

led me to nowhere.

*The problem might be easy, but I study by myself and have no one to consult, I appreciate

any tips or hints, thanks in advance.
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tiny-tim
#2
May21-12, 06:04 PM
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Hello Ammar Kurd! Welcome to PF!

(try using the X2 button just above the Reply box )
Quote Quote by Ammar Kurd View Post
Verify that y = (x2/2) + ((x/2)*√(x2 + 1)) + ln(√(x+√(x2 +1))) is a solution of the

equation 2y = x*y' + ln(y').....(1)
Show us what you got for y'
Ammar Kurd
#3
May22-12, 05:06 AM
P: 3
Thank you for the replay, I got

y' = x + 0.5√(x2+1) + (x2 / (2√(x2+1))) + (x+√(x2+1) / (x√(x2+1)+x2+1))

I also tried to simplify the (y) in this way:

y = 0.5x2 + 0.5x√(x2+1) + ln(√(x+√(x2+1)))

= .5x(x+√(x2+1)) + 0.5ln(x+√(x2+1))

then putting G(x) = x+√(x2+1)

y becomes:

y = 0.5x*G(x) + 0.5ln(G(x)), and

y' = 0.5*G(x) +0.5xG'(x) + 0.5*(G'(x)/G(x))

But when I substitute in the differential equation it only get complicated...

*Thank you for the x2 tip .

tiny-tim
#4
May22-12, 06:08 AM
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Need help with differential equation.

Hello Ammar Kurd!
Quote Quote by Ammar Kurd View Post
y = (x^2/2) + ((x/2)*√(x^2 + 1)) + ln(√(x+√(x^2 +1)))
Quote Quote by Ammar Kurd View Post
Thank you for the reply, I got

y' = x + 0.5√(x2+1) + (x2 / (2√(x2+1))) + (x+√(x2+1) / (x√(x2+1)+x2+1))
Yes, that's correct, except I think there should be a factor 2 in the last term.

I don't know how you got that last term, but it simplifies, to 1/√(x2+1)
Ammar Kurd
#5
May22-12, 10:09 AM
P: 3
Quote Quote by tiny-tim View Post
I don't know how you got that last term, but it simplifies, to 1/√(x2+1)
That was my problem I didn't notice that the last term can be further simplified .

Thank you, Problem solved


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