Initial Value Problems

Solve the initial value problems in Exercises 71–90.

dv/dt = (1/2)sec t tan t, v(0) = 1

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Identify the given differential equation and initial condition: \(\frac{dv}{dt} = \frac{1}{2} \sec t \tan t\), with \(v(0) = 1\).

Recognize that this is a separable differential equation where the right side is a function of \(t\) only, so you can integrate both sides with respect to \(t\) to find \(v(t)\).

Set up the integral: \(v(t) = \int \frac{1}{2} \sec t \tan t \, dt + C\), where \(C\) is the constant of integration.

Recall the integral formula: \(\int \sec t \tan t \, dt = \sec t + C\). Use this to integrate the right side.

Apply the initial condition \(v(0) = 1\) to solve for the constant \(C\) by substituting \(t=0\) and \(v=1\) into the expression for \(v(t)\).

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주요 개념

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

Differential Equations

A differential equation relates a function with its derivatives. Solving it involves finding the original function that satisfies the given relationship between the function and its rate of change.

Initial Value Problems (IVP)

An initial value problem specifies the value of the unknown function at a particular point, allowing for a unique solution to the differential equation by applying this initial condition.

Integration of Trigonometric Functions

Solving the differential equation requires integrating trigonometric expressions like sec t and tan t. Knowing standard integrals and techniques for these functions is essential to find the explicit solution.

Initial Value ProblemsSolve the initial value problems in Exercises 71–90.dv/dt = (1/2)sec t tan t, v(0) = 1