Solve the differential equation using the method of variation of parameters.
Select the correct answer.
$$y ^ { tt } - y ^ { t } = e ^ { 2 x }$$
A) $y ( x ) = c _ { 1 } + c _ { 2 } e ^ { x } + \frac { e ^ { 2 x } } { 2 }$
B) $y ( x ) = c _ { 1 } + c _ { 2 } x e ^ { 2 x }$
C) $y ( x ) = c _ { 1 } + x e ^ { 2 x }$
D) $y ( x ) = c _ { 1 } + c _ { 2 } x e ^ { x } + \frac { e ^ { 2 x } } { 2 }$
E) $y ( x ) = c _ { 1 } + ( 1 + x ) x e ^ { 2 x }$

Question

Quizplus · Accepted Answer

The complementary solution is $y_c = c_1 + c_2 e^x$. Using variation of parameters, the particular solution is $\frac{e^{2x}}{2}$. Thus, the general solution is $y(x) = c_1 + c_2 e^x + \frac{e^{2x}}{2}$.

Solve the Differential Equation Using the Method of Variation of Parameters