 Margie Marx

2021-12-19

Solve the given differential equation by undetermined coefficients
$y{}^{″}-{14}^{\prime }+49y=35x+4$ Ana Robertson

Expert

Consider the differential equation
...(1)
Rewrite the equation,$\left({D}^{2}-14D+49\right)y=30$ where
Auxiliary equation is,f(m)=0
${m}^{2}-14m+49=0$
$\left(m-7{\right)}^{2}=0$
$m=7,7$
Hence the complementary solution is,
${y}_{c}={c}_{1}{e}^{7x}+{c}_{2}{e}^{7x}$ otoplilp1

Expert RizerMix

Expert

no, here is the continuation of the solution
Apply the undetemined coefficient method to find the particular solution as follows:
${y}_{p}=Ax+B$
Find the first and second derivatives,
${y}_{p}\text{'}=A$,
${y}_{p}\text{'}\text{'}=0$
Substitute these values in equation $\left(1\right)$
$0-14A+49\left(Ax+B\right)=35x+4$
$-14A+49Ax+49B=35x+4$
$\left(-14A+49B\right)+49Ax=35x+4$
Compare the coefficients of constant and x terms,
$49A=35$
$-14A+49B=4$
Solve the above two equations,
$A=\frac{5}{7}$ and $B=\frac{2}{7}$

$\left(A=\left(35\right)/\left(49\right)\right),\left(-14\left(5/7\right)+49B=4\right),\left(-10+49B=4\right),\left(49B=14\right),\left(B=2/7\right):$
Hence the particular solution is,
${y}_{p}=\frac{5}{7}x+\frac{2}{7}$
The general solution is,
$y={y}_{c}+{y}_{p}$
${y}_{c}={c}_{1}{e}^{7x}+{c}_{2}{e}^{7x}+\frac{5}{7}x+\frac{2}{7}$
Therefore, the required general solution is ${y}_{c}={c}_{1}{e}^{7x}+{c}_{2}{e}^{7x}+\frac{5}{7}x+\frac{2}{7}$

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