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**algorithm** Hello,

Thank you for the help.

This is another ODE:

$\displaystyle y'' + y' = x^{2}$,

conditions: $\displaystyle y(0) = 0, y'(0) = 0 $

Complementary function, y_c:

$\displaystyle m^{2} + m = 0$

$\displaystyle m = 0, -1$

$\displaystyle y_c = (C1 + C2x)e^{-x}$

Particular integral y_p:

Try $\displaystyle y_p = Ax^{2} + Bx + C$ this C is part of your complimentary solution

$\displaystyle y_p' = 2Ax + B$

$\displaystyle y_p'' = 2A$

$\displaystyle => 2A + 2Ax + B = x^{2}$

$\displaystyle 2A + B = 0$, and $\displaystyle 2A = 0$, giving $\displaystyle A = B = 0$.

$\displaystyle y = C1 + C2e^{-x}$

$\displaystyle y' = -C2e^{-x}$

$\displaystyle 0 = C1 + C2$

$\displaystyle 0 = -C2$

$\displaystyle C1 = C2 = 0$, giving the final solution:

$\displaystyle y = 0$

The correct answer has more $\displaystyle x$'s.

Thank you