Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

Help with a partial differential equation question?

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  • 1 month ago

    I'll give it a shot......

    u_xx - xt^2 = 0       u_xx = xt^2      u_x  = ∫ xt^2 dx  =  (1/2)x^2 t^2 + C1

      where C1 = f(t)

    u = ∫  [  (1/2)x^2 t^2 + C1 ]dx   =   (1/6) x^3  t^2  + ( C1 ) x  +  C2     

    where C2 = g(t)

    Now  boundary conditions..... 

    u(0,t)  =  0  + 0 + C2  =  t^2 + 1      so    C2 = t^2 + 1

    u(1,t)  =  (1/6)t^2  +  C1 +  t^2 + 1   =  (4/3)t^2 

     C1 = (4/3)t^2  -   (1/6)t^2  -  [  t^2 + 1  ] 

    C1  = (1/6)t^2 - 1

    u(x,t)  = (1/6) x^3  t^2   +  ( (1/6)t^2 - 1  )x  +  ( t^2 + 1 )

    double checking... u_x = (1/2)x^2 t^2 +  (1/6)t^2 - 1

    u_xx = x t^2 + 0       so u_xx - xt^2 = 0      :-)      I think this works !!!

    It's been awhile since I've done  PDE  but I used to solve them all the time before !!!          don't forget to choose a Best Answer.....

  • 1 month ago

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