Solve differential equation y=e^(6x)−3x−2

Solve differential equation $y={e}^{\left(}6x\right)-3x-2$
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Luvottoq

${y}^{\prime }-6y=7x+3$
$y={e}^{6x}-3x-2$
${y}^{\prime }\left(x\right)+p\left(x\right)y=q\left(x\right)$
$p\left(x\right)=6$
In order to solve the ordinary first order differential equation, we use integration factor that is:
$\mu \left(x\right)={e}^{\int p\left(x\right)dx}$
$\mu \left(x\right)={e}^{\int \left(-6\right)dx}$
$\mu \left(x\right)={e}^{-6\int 1dx}$
$\mu \left(x\right)={e}^{-6x}$
Write the equation in the form
$\left(\mu \left(x\right)y{\right)}^{\prime }=\mu \left(x\right)q\left(x\right)$

So
$\left({e}^{-6x}y{\right)}^{\prime }={e}^{-6x}\left(7x+3\right)$
$\left({e}^{-6x}y{\right)}^{\prime }=6x{e}^{-6x}+3{e}^{-6x}$
Solve
$\left({e}^{-6x}y{\right)}^{\prime }=7x{e}^{-6x}+3{e}^{-6x}$
${e}^{-6x}y=7/36\left(-6{e}^{-6x}x-{e}^{-6x}\right)+3\left(-1/6{e}^{-6x}\right)+c$
${e}^{-6x}y=42/36{e}^{-6x}x-7/36{e}^{-6x}-3/6{e}^{-6x}+c$
${e}^{-6x}y=-42/36{e}^{-6x}x-7/36{e}^{-6x}-18/36{e}^{-6x}+c$
${e}^{-6x}y=\left(-42{e}^{-6x}-25{e}^{-6x}\right)/36+c$
$y=-42{e}^{-6x}x-25{e}^{-6x}/\left(36{e}^{-6x}\right)+c$