Use power series to solve the differential equation.
$$y ^ { t} = 4 x y$$
A)
$$y ( x ) = c e ^ { - \frac { 4 x ^ { 2 } } { 2 } }$$
B)
$$y ( x ) = c e ^ { \frac { 4 x ^ { 2 } } { 2 } }$$
C)
$$y ( x ) = e ^ { - \frac { x ^ { 2 } } { 2 } }$$
D) $y ( x ) = c e ^ { 4 x }$
E)
$$y ( x ) = 4 e ^ { \frac { x ^ { 2 } } { 2 } }$$

Question

Quizplus · Accepted Answer

The differential equation is a first-order linear differential equation. We can solve it by using the integrating factor method. The integrating factor is $e^{\int 4x dx} = e^{2x^2}$. Multiplying both sides of the differential equation by the integrating factor, we get$$e^{2x^2}y' - 4xe^{2x^2}y = 0.$$Now, we can integrate both sides of the equation to get$$y e^{2x^2} = C,$$where $C$ is a constant. Dividing both sides of the equation by $e^{2x^2}$, we get$$y = Ce^{-2x^2}.$$Therefore, the solution to the differential equation is $y(x) = Ce^{-2x^2}$, which is choice B.

Use Power Series to Solve the Differential Equation yt=4xyy ^ { t} = 4 x yyt=4xy

Use Power Series to Solve the Differential Equation $y ^ { t} = 4 x y$