Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$\begin{array} { l }
y ^ { \prime } + y = e ^ { 2 x } \
y = 5 e ^ { - x } + \frac { 1 } { 3 } e ^ { 2 x }
\end{array}$

Question

Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-$\begin{array} { l } 
y ^ { \prime } + y = e ^ { 2 x } \
y = 5 e ^ { - x } + \frac { 1 } { 3 } e ^ { 2 x }
\end{array}$

Quizplus · Accepted Answer

To check if the given function y = f(x) is a solution of the differential equation y' + y = e^(2x), we need to substitute y = f(x) into the differential equation and see if it is satisfied.

y' + y = e^(2x)
f'(x) + f(x) = e^(2x)

Now we differentiate the given function y = 5e^(-x) + (1/3)e^(2x) to get:

f(x) = 5e^(-x) + (1/3)e^(2x)
f'(x) = -5e^(-x) + (2/3)e^(2x)

Substituting these into the differential equation, we get:

f'(x) + f(x) = (-5e^(-x) + (2/3)e^(2x)) + (5e^(-x) + (1/3)e^(2x)) = e^(2x)

Thus, the given function y = f(x) satisfies the differential equation y' + y = e^(2x). Therefore, the answer is A.

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