33–38. {Use of Tech} Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem.
z(x) = (z² + 4)/(x² + 16), z(4) = 2

Verified step by step guidance

1. Rewrite the given differential equation in the form \( \frac{dz}{dx} = \frac{z^2 + 4}{x^2 + 16} \) to clearly identify the derivative \( \frac{dz}{dx} \).
2. Separate the variables by rearranging terms to isolate \( z \) on one side and \( x \) on the other: \( \frac{dz}{z^2 + 4} = \frac{dx}{x^2 + 16} \).
3. Integrate both sides: compute \( \int \frac{dz}{z^2 + 4} \) and \( \int \frac{dx}{x^2 + 16} \). Recall that \( \int \frac{du}{u^2 + a^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \).
4. After integrating, write the implicit solution by equating the two antiderivatives plus a constant of integration \( C \): \( \frac{1}{2} \arctan\left(\frac{z}{2}\right) = \frac{1}{4} \arctan\left(\frac{x}{4}\right) + C \).
5. Use the initial condition \( z(4) = 2 \) to solve for the constant \( C \), then express the implicit solution including this constant. This implicit equation represents the solution to the initial value problem.

Key Concepts

Implicit Solutions of Differential Equations

An implicit solution is a relation involving both the dependent and independent variables that satisfies the differential equation without explicitly solving for the dependent variable. Such solutions may define multiple functions, and additional conditions help identify the specific solution relevant to the initial value problem.

Initial Value Problems (IVPs)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition allows us to determine the unique solution curve among many possible implicit or explicit solutions.

Use of Technology for Graphing Solutions

Graphing software helps visualize implicit solutions that are difficult to solve explicitly. By plotting the implicit relation and applying the initial condition, one can identify the correct solution branch and better understand the behavior of the solution.