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Math Help - Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

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    Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

    Hello guys any help would be appreciated for this question.

    Show that the general solution of y=x*p + f(p) is y=cx+f(c), where p=dy/dx and c is an arbitrary constant. Also show that there is a 2nd solution obeying the diff. equation d/dp(f(p)) + x =0. Finally find the singular solution obeying both of the diff equations.


    Obviously the first equation is not a separable equation. Also can't use the integrating factor since we have a function of the derivative in our equation. Can anyone tell me how do we solve this type of differential equations since is the first time I see this type? By direct differentiation of the given solution is easy to show that it is a solution but I don't think so this is a way. And after solving the first equation how do I proceed for the other parts?

    Thanks in advance for any help!
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  2. #2
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    re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

    That is actually the way to solve this ODE. If you differentiate wrt x you get

    p = p + xp' + f'(p)p'

    or

    p'\left( f'(p) + x\right) = 0 - two cases

    (1)  p ' = 0 gives y = ax + b (subs this back into your original ODE to find conditions on a and b)

    (2) f'(p) +x = 0 is the second choice
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    Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

    Thank you very much! Substituting in the original ODE I get b=f(a), so from the first moment should I have put that y=cx+b and then find that b=f(c)? Also how do I find a singular solution obeying the two equations (y=x*p + f(p), d/dp (f(p)) + x= 0)

    Thanks again very much!
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    Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

    I am still stucked with this exercise, can anyone please help me about how do I find a singular solution obeying the two equations (y=x*p + f(p), d/dp (f(p)) + x= 0)? Thank you very much!
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  5. #5
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    Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

    Suppose that f(p) is some know function then

    x = -f'(p)

    and from the first

    y = p x + f(p) = f(p) - p f'(p).

    So now you have x and y in terms of the parameter  p (i.e. you have a parametric solution).
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    Re: Show solution of y=x*p + f(p) is y=cx+f(c), with p = dy/dx

    thank you very much, your help is appreciated!
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