# If y=xe^7 is a solution of a linear homogeneous 2nd order DE, then another solution might be

If $y=x{e}^{7x}$ is a solution of a linear homogeneous 2nd order DE, then another solution might be?
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Damarion Pierce
For a second order linear homogenous equations,
If roots of auxiliary equation are
1. real and distinct then the solution is
$y={C}_{1}{e}^{{m}_{1}x}+{C}_{2}{e}^{{m}_{2}x}$
2.real and equal then the solution is
$y={C}_{1}{e}^{mx}+{C}_{2}x{e}^{mx}$
3.complex then the solution is
$y={e}^{\alpha x}\left({C}_{1}\mathrm{cos}\left(\beta x\right)+{C}_{2}\mathrm{sin}\left(\beta x\right)\right)$
Given $y=x{e}^{7x}$ is the solution
the solution is similar for the case of real and equal roots
therefor the other solution is
$y={e}^{7x}$