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

검증된 단계별 안내

Rewrite the given differential equation in the form \( \frac{dz}{dx} = \frac{z^2 + 4}{x^2 + 16} \) to clearly identify the derivative \( \frac{dz}{dx} \).

Separate the variables by rearranging terms to isolate \( z \) on one side and \( x \) on the other: \( \frac{dz}{z^2 + 4} = \frac{dx}{x^2 + 16} \).

Integrate both sides: compute \( \int \frac{dz}{z^2 + 4} \) and \( \int \frac{dx}{x^2 + 16} \). Recall that \( \int \frac{du}{u^2 + a^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \).

After integrating, write the implicit solution by equating the two antiderivatives plus a constant of integration \( C \): \( \frac{1}{2} \arctan\left(\frac{z}{2}\right) = \frac{1}{4} \arctan\left(\frac{x}{4}\right) + C \).

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.