Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$\begin{array} { l }
y ^ { \prime } - y = 5 e ^ { x } \
y = e ^ { x } + 5 x e ^ { x }
\end{array}$

Question

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$\begin{array} { l } 
y ^ { \prime } - y = 5 e ^ { x } \
y = e ^ { x } + 5 x e ^ { x }
\end{array}$

Quizplus · Accepted Answer

Firstly, let's take the derivative of the given function $y=e^x+5xe^x$:
$$y'=e^x+5e^x+5xe^x$$
Now substitute this and the given function $y=e^x+5xe^x$ into the differential equation:
\begin{align*}
y'-y&=5e^{x}\
(e^x+5e^x+5xe^x)-(e^x+5xe^x)&=5e^x\
5e^x&=5e^x
\end{align*}
Since the left-hand side equals the right-hand side, we can conclude that the given function is indeed a solution to the differential equation. Therefore, the answer is A)True.

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