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

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

i dont know and i dont care for math bc it is a stupied subject

step-by-step explanation:

i think the correct option is the second option.