# Need help with differential equation.

by Ammar Kurd
Tags: diff equation, problem, verify that
 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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Hello Ammar Kurd! Welcome to PF!

(try using the X2 button just above the Reply box )
 Quote by Ammar Kurd 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'
 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 .
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## Need help with differential equation.

Hello Ammar Kurd!
 Quote by Ammar Kurd y = (x^2/2) + ((x/2)*√(x^2 + 1)) + ln(√(x+√(x^2 +1)))
 Quote by Ammar Kurd 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)
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 Quote by tiny-tim 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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