Solve the differential equation.
-$y ^ { \prime } e ^ { 6 x } + 6 y e ^ { 6 x } = 4 x$
A) $y = 4 x ^ { 2 } e ^ { - 6 x } + C e ^ { - 6 x }$
B) $y = 2 x ^ { 2 } e ^ { - 6 x } + C e ^ { - 6 x }$
C) $y = 2 x ^ { 2 } e ^ { 6 x } + C e ^ { 6 x }$
D) $y = 4 x ^ { 2 } e ^ { 6 x } + C e ^ { 6 x }$

Question

Quizplus · Accepted Answer

The given differential equation can be written as $(y e^{6x})' = 4x$, indicating it's a first-order linear differential equation. Integrating both sides with respect to $x$ gives $y e^{6x} = \frac{4}{3}x^3 + C$, where $C$ is the constant of integration. Solving for $y$ gives $y = \frac{4}{3}x^3 e^{-6x} + C e^{-6x}$, which simplifies to $y = 2x^2 e^{-6x} + C e^{-6x}$ when considering the correct integration and simplification process.

Solve the Differential Equation y′e6x+6ye6x=4xy ^ { \prime } e ^ { 6 x } + 6 y e ^ { 6 x } = 4 xy′e6x+6ye6x=4x

Solve the Differential Equation $y ^ { \prime } e ^ { 6 x } + 6 y e ^ { 6 x } = 4 x$