# Solve the given differential equation by undetermined coefficients y''+7y'+6y=30

Solve the given differential equation by undetermined coefficients
$y{}^{″}+7{y}^{\prime }+6y=30$
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Stella Calderon
Consider the differential equation
$y{}^{″}+7{y}^{\prime }+6y=30$...(1)
Rewrite the equation,$\left({D}^{2}+7D+6\right)y=30$ where $D=\frac{d}{dx}$
Auxiliary equation is,
$f\left(m\right)=0$
${m}^{3}+7m+6=0$
${m}^{2}+6m+m+6=0$
$m\left(m+6\right)+1\left(m+6\right)=0$
$\left(m+6\right)\left(m+1\right)=0$
$m=-1,-6$
Hence the complementary solution is,
${y}_{c}={c}_{1}{e}^{-x}+{c}_{2}{e}^{-6x}$
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Medicim6
Can I get full answer, please. I think tha is not full.
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RizerMix

Oh, sure, sorry
Apply the undetemined coefficient method to find the particular solution as follows:
${y}_{p}=Ax+B$
Find the first and second derivatives,
${y}_{p}^{\prime }=A,$
${y}_{p}^{″}=0$
Substitute these values in equation (1)
$0+7A+6\left(Ax+B\right)=30$
$\left(7A+6B\right)+6Ax=30$
Compare the coeficients of constant and x terms,
$6A=0$
$7A+7B=0$
Solve the above two equations,

Hence the particular solution is,
${y}_{p}=0\left(x\right)+5$
$=5$
The general solution is,
$y={y}_{c}+{y}_{p}$
${y}_{c}={c}_{1}{e}^{-x}+{c}_{2}{e}^{-6x}+5$
Therefore, the required general solution is ${y}_{c}={c}_{1}{e}^{-x}+{c}_{2}{e}^{-6x}$