2022-03-22

Solve the equation
$y{}^{″}-4y=2+3x-{x}^{2}$

### Answer & Explanation

etsahalen5tt

Given differential equation is
$y{}^{″}-4y={x}^{2}+3x+2$
This is a second order linear non-homogeneous equation of the form of
$ay{}^{″}+b{y}^{\prime }+cy=g\left(x\right)$
General solution of this equation is
$y={y}_{h}+{y}_{p}$
Where
${y}_{h}$ is solution of $ay{}^{″}+b{y}^{\prime }+cy=0$
And ${y}_{p}$ is the solution for any function that satisfies non-homogeneous equation
Hense,
${y}_{h}={c}_{1}{e}_{1}^{m}+{c}_{2}{e}_{2}^{m}$
Where ${m}_{1}$ and ${m}_{2}$ are the roots of
Auxiliary equation of $y{}^{″}-4y$ is
${m}^{2}-4=0$

Then ${y}_{h}={c}_{1}{e}^{2}+{c}_{2}{e}^{-2}$
Let $y=A{x}^{2}+Bx+C$
$y=2Ax+B$
$y=2A$
Similarly
$B=\frac{-3}{4}$
And
$\left(2A-4C\right)=2$
$⇒2\cdot \frac{1}{4}-4C=2$
$⇒\frac{1}{2}-4C=2$
$C=\frac{-3}{8}$
Thus solution is
$y={c}_{1}{e}^{2x}+{c}_{2}{e}^{-2x}+\frac{{x}^{2}}{4}-\frac{3x}{4}-\frac{3}{8}$

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