Question

Show that the given general solution satisfies the given differential equation, showing all work carefully. Do NOT do this by solving the given differential equation, but rather prove that the whole general solution makes the equation true.
DE= x^2y''- 8xy +8y=16
Gen. Sol y=c1x+c2x^8+2

Answer 1

Explanation:

Differentiating the solution, we have ...

  y' = c1 +8c2x^7

  y'' = 56c2x^6

Putting this into the DE, we have ...

  x^2y'' -8xy' +8y = 16 . . . . . . . different from your problem statement

  x^2(56c2x^6) -8x(c1 +8c2x^7) +8(c1x +c2x^8 +2) = 16

  56c2x^8 -8c1x -64c2x^8 +8c1x +8c2x^8 +16 = 16

  x^8(56c2 -64c2 +8c2) +x(-8c1 +8c1) +16 = 16

  0 +16 = 16 . . . . QED