Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $y ^ { \prime } = e ^ { - 7 x } - 7 y$
Initial condition: $y ( 0 ) = 0$
Solution candidate: $y = 7 xe ^ { - 7 x }$

Question

Quizplus · Accepted Answer

To verify if $y = 7xe^{-7x}$ is a solution, we differentiate it to get $y' = 7e^{-7x} - 49xe^{-7x}$ and compare it to the given differential equation $y' = e^{-7x} - 7y$. Substituting $y = 7xe^{-7x}$ into the right side of the differential equation gives $e^{-7x} - 7(7xe^{-7x}) = e^{-7x} - 49xe^{-7x}$, which does not match the derived $y'$, indicating the proposed solution does not satisfy the differential equation.

Determine If the Given Function Y = F(x) Is a Solution