Chapter 7.3, Problem 29E

To determine

To solve the differential equation: $\frac{dy}{dx}=x^2{e}^{4x}$

Answer to Problem 29E

The solution of the differential equation $\frac{dy}{dx}=x^2{e}^{4x}$ is $e^{4x}\left[\frac{x^2}{4}-\frac{x}{8}+\frac{1}{32}\right]+C$

Explanation of Solution

Given information: $\frac{dy}{dx}=x^2{e}^{4x}$

Formula used: ${\displaystyle \int uvdx=u{\displaystyle \int vdx-{\displaystyle \int \left(u^{\prime }{\displaystyle \int vdx}\right)}}}dx$

Calculation :

The given equation becomes as:

  $dy=x^2{e}^{4x}dx$

Integrating above equation with respect to $x$ , the expression becomes as:

  $y={\displaystyle \int x^2{e}^{4x}}dx+C$

  $=\left[(x^2)\left(\frac{e^{4x}}{4}\right)-(2x)\left(\frac{e^{4x}}{16}\right)+(2)\left(\frac{e^{4x}}{64}\right)\right]+C$

  $=e^{4x}\left[\frac{x^2}{4}-\frac{x}{8}+\frac{1}{32}\right]+C$

Hence, the solution of the differential equation is $e^{4x}\left[\frac{x^2}{4}-\frac{x}{8}+\frac{1}{32}\right]+C$