Need help with differential equation.

  • Thread starter Ammar Kurd
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  • #1
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.
 

Answers and Replies

  • #2
tiny-tim
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Welcome to PF!

Hello Ammar Kurd! Welcome to PF! :smile:

(try using the X2 button just above the Reply box :wink:)
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' :smile:
 
  • #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 :smile:.
 
  • #4
tiny-tim
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Hello Ammar Kurd! :smile:
y = (x^2/2) + ((x/2)*√(x^2 + 1)) + ln(√(x+√(x^2 +1)))
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. :smile:

I don't know how you got that last term, but it simplifies, to 1/√(x2+1) :wink:
 
  • #5
I don't know how you got that last term, but it simplifies, to 1/√(x2+1) :wink:

That was my problem I didn't notice that the last term can be further simplified :redface:.

Thank you, Problem solved :smile:
 

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