# Find the particular solution of a non-homogeneous equation using Laplace transformation, y(0) = 0, y'(0)=0, y"(0)=0 y'''-5y''-22y'+56y=12-32e^{-8x}+2e^{4x}

Find the particular solution of a non-homogeneous equation using Laplace transformation,
$y\left(0\right)=0,$
${y}^{\prime }\left(0\right)=0,$
$y"\left(0\right)=0$
${y}^{‴}-5{y}^{″}-22{y}^{\prime }+56y=12-32{e}^{-8x}+2{e}^{4x}$
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Step 1
Given non homogenous equation
${y}^{‴}-5{y}^{″}-22{y}^{\prime }+56y=12-32{e}^{-8x}+2{e}^{4x}$
With initial conditions $y\left(0\right)=0,{y}^{\prime }\left(0\right)=0,y"\left(0\right)=0$
Taking Laplace transform on both sides,
$L\left\{{y}^{‴}\right\}-5L\left\{{y}^{″}\right\}-22L\left\{{y}^{\prime }\right\}+56L\left\{y\right\}=L\left\{12\right\}-32L\left\{{e}^{-8x}\right\}+2L\left\{{e}^{4x}\right\}$
${s}^{3}\overline{y}-5{s}^{2}\overline{y}-22s\overline{y}+56\overline{y}=\frac{12}{s}-\frac{32}{\left(s+8\right)}+\frac{2}{\left(s-4\right)}$
$\overline{y}\left({s}^{3}-5{s}^{2}-22s+56\right)=\frac{12}{s}-\frac{32}{\left(s+8\right)}+\frac{2}{\left(s-4\right)}$
$\overline{y}\left(s-2\right)\left(s+4\right)\left(s-7\right)=\frac{12}{s}-\frac{32}{\left(s+8\right)}+\frac{2}{\left(s-4\right)}$
$\overline{y}=\frac{12}{s\left(s-2\right)\left(s+4\right)\left(s-7\right)}-\frac{32}{\left(s+8\right)\left(s-2\right)\left(s+4\right)\left(s-7\right)}+\frac{2}{\left(s-4\right)\left(s-2\right)\left(s+4\right)\left(s-7\right)}$
Dividing each term on right into partial fraction,
$\overline{y}=\left(\frac{3}{14s}-\frac{1}{5\left(s-2\right)}-\frac{1}{22\left(s+4\right)}+\frac{12}{385\left(s-7\right)}\right)-\left(-\frac{4}{75\left(s+8\right)}-\frac{8}{75\left(s-2\right)}+\frac{1}{33\left(s+4\right)}+\frac{32}{825\left(s-7\right)}\right)+\left(-\frac{1}{24\left(s-4\right)}+\frac{1}{30\left(s-2\right)}-\frac{1}{264\left(s+4\right)}+\frac{2}{165\left(s-7\right)}\right)$
Taking inverse Laplace transform on both sides,
$y\left(x\right)=\frac{3}{14}-\frac{1}{5}{e}^{2x}-\frac{1}{22}{e}^{-4x}+\frac{12}{385}{e}^{7x}+\frac{4}{75}{e}^{-8x}+\frac{8}{75}{e}^{2x}-\frac{4}{33}{e}^{-4x}-\frac{32}{825}{e}^{7x}-\frac{1}{24}{e}^{4x}+\frac{1}{30}{e}^{2x}-\frac{1}{264}{e}^{-4x}+\frac{2}{165}{e}^{7x}$