Question

Solve the differential equation by method of undetermined cooefficient..
d^2y/dx^2=9x^2+2x-1

Answer 1

Is the equation\[y''=9 x^2+2 x-1\] ?

Answer 2

Yes.

Answer 3

Refer to the attachment a solution by WolframAlpha.

Answer 4

Homogeneous solution:
\(y''=0\) yields the characteristic equation \(r^2=0\), giving you the root \(r=0\) with multiplicity two.

The homogeneous solution is thus \(y_c=C_1+C_2x\).

Non-homogeneous solution:
Normally, as a guess, you would try \(y_p=Ax^2+Bx+C\), but since the right hand side only contains \(y''\), you must multiply the guess by \(x^2\). This is so that you have like terms when solving for the undetermined coefficients.
\[~~~~~~~~~~~~~\begin{align*}
y_p&=Ax^4+Bx^3+Cx^2\\
y_p~'&=4Ax^3+3Bx^2+2Cx\\
y_p~''&=12Ax^2+6Bx+2C
\end{align*}\]
Substitute into the diff eq:
\[\begin{align*}
y''&=9x^2+2x-1\\
12Ax^2+6Bx+2C&=9x^2+2x-1
\end{align*}\]
Now solve for the coefficients:
\[\begin{cases}
12A=9\\ 6B=2\\ 2C=-1
\end{cases}\]