Combine the terms y and $y ^ { \prime }$ into the derivative of a product, then solve the equation. $\mathrm { e } ^ { 3 \mathrm { t } ^ { 2 } } \mathrm { y } ^ { \prime } + 6 \mathrm { te } ^ { 3 \mathrm { t } ^ { 2 } } \mathrm { y } = \frac { 5 \sqrt { \mathrm { t } } } { 4 }$. Is this the solution: $y = \frac { 5 } { 6 } t ^ { 3 / 2 } e ^ { - 3 t ^ { 2 } } + C e ^ { - 3 t ^ { 2 } } ?$

Question

Quizplus · Accepted Answer

The given equation is a first-order linear differential equation that can be solved using an integrating factor. The integrating factor is $e^{3t^2}$, which transforms the left side into the derivative of a product: $\frac{d}{dt}(e^{3t^2}y)$. Solving this gives the solution $y = \frac{5}{6}t^{3/2}e^{-3t^2} + Ce^{-3t^2}$, which matches the proposed solution.

Combine the Terms Y And y′y ^ { \prime }y′ Into the Derivative of a Product, Then Solve the Equation

Combine the Terms Y And $y ^ { \prime }$ Into the Derivative of a Product, Then Solve the Equation