Question

In Exercises \(11-16,\) solve the initial value problem. Confirm your answer by checking that it conforms to the slope field of the differential equation. $$\frac{d z}{d x}=x^{3} \ln x\( and \)z=5\( when \)x=1$$

Short answer

The solution to the differential equation is \(z = \frac{x^{4} \ln x}{4} - \frac{x^{4}}{16} + \frac{81}{16}\).

Step-by-step solution

- Separation of Variables

Rearrange the ODE to separate variables. We want \(z\) and \(dz\) on one side of the equation and \(x\) and \(dx\) on the other. So, we have \(dz = x^{3} \ln x \, dx\).

- Integration of Both Sides

Integrate both sides of the equation. The left side is easily integrated as \(z\). The right side is a bit trickier: it can be integrated by parts, using the formula: \(\int u \, dv = uv - \int v \, du\), where \(u = \ln x\) and \(dv = x^{3} dx\). This leads to: \[z = \frac{x^{4} \ln x}{4} - \frac{x^{4}}{16} + C\], where \(C\) is the constant of integration.

- Apply the Initial Condition

Substitute the initial conditions into the resulting equation and solve for \(C\). So if \(x = 1\) and \(z = 5\), we get \[5 = \frac{1^{4} \ln 1}{4} - \frac{1^{4}}{16} + C\], solving for \(C\) we find \(C = \frac{81}{16}\). Therefore, our solution is \(z = \frac{x^{4} \ln x}{4} - \frac{x^{4}}{16} + \frac{81}{16}\).

- Confirming the Solution

At this point, you would normally plot the slope field of the original differential equation and superimpose the function just obtained to confirm this is a correct solution visually. However, this process is tedious to do by hand and generally resort to using a mathematical software like MATLAB, Python or a graphing calculator.