Question

In Exercises \(29-32,\) solve the differential equation. $$\frac{d y}{d x}=x^{2} e^{4 x}$$

Short answer

The solution to the given differential equation is \(y(x) = 0.25 x^{2} e^{4 x} - 0.125 x e^{4 x} + 0.0625 e^{4 x} + C\).

Step-by-step solution

Identify the problem

We are given a first-order differential equation of the form \(\frac{d y}{d x}=x^{2} e^{4 x}\). Our goal is to find the function \(y(x)\) that satisfies this equation.

Integrate the Right Side by Parts

The right side of the equation looks like it needs to be integrated by parts. Recall the integration by parts formula: \(\int u dv = uv - \int v du\). We need to apply this twice here. Let's start by setting \(u = x^{2}\) and \(dv = e^{4 x} dx\). Computing the differential, we have \(du = 2x dx\) and integrating \(dv\), we obtain \(v = 0.25 e^{4 x}\).

Apply the Integral by Parts Formula

Substituting into the formula, we obtain \(x^{2} e^{4 x} = u v - \int v du = x^{2} (0.25 e^{4 x}) - \int 0.25 e^{4 x} (2x dx)\). Let's simplify that to \(x^{2} e^{4 x} = 0.25 x^{2} e^{4 x} - 0.5 \int x e^{4 x} dx\). Note that for the new integral, you should again use integration by parts with \(u = x\) and \(dv = e^{4 x} dx\).

Integrate by Parts Once More

Again let's compute \(du = dx\), and \(v = 0.25 e^{4 x}\). Substituting into the formula, we get \(x e^{4 x} = x (0.25 e^{4 x}) - \int 0.25 e^{4 x} dx = 0.25 x e^{4 x} - 0.25 \int e^{4 x} dx\). Integrating \(e^{4x}\) yields \(0.25 e^{4x}\), so the integral becomes \(= 0.25 x e^{4 x} - 0.25 (0.25 e^{4 x})\).

Substitute Back

Substitute the results from step 3 and step 4 back into the original integration, to get \(\int x^{2} e^{4 x} dx = 0.25 x^{2} e^{4 x} - 0.5 \times (0.25 x e^{4 x} - 0.25 \times 0.25 e^{4 x}) = 0.25 x^{2} e^{4 x} - 0.125 x e^{4 x} + 0.0625 e^{4 x}\).

Integrate the Left Side

The derivative of \(y\) with respect to \(x\) was given, which means integrating it will yield the original function \(y\). So, \(\int dy = y\).

Equate and Solve

We equate the integration results of the left and right side. Therefore, the solution of the differential equation is \(y(x) = 0.25 x^{2} e^{4 x} - 0.125 x e^{4 x} + 0.0625 e^{4 x} + C\), where \(C\) represents the constant of integration.