In Exercises 23–28, solve the initial value problem.
x dy + (y - cos x) dx = 0, y(π/2) = 0

검증된 단계별 안내

Rewrite the given differential equation in the form \( M(x,y) \, dx + N(x,y) \, dy = 0 \). Here, the equation is \( x \, dy + (y - \cos x) \, dx = 0 \), so rearranged as \( (y - \cos x) \, dx + x \, dy = 0 \).

Check if the differential equation is exact by computing the partial derivatives \( \frac{\partial M}{\partial y} \) and \( \frac{\partial N}{\partial x} \), where \( M = y - \cos x \) and \( N = x \). If they are equal, the equation is exact.

If the equation is exact, find the potential function \( \Psi(x,y) \) such that \( \frac{\partial \Psi}{\partial x} = M = y - \cos x \) and \( \frac{\partial \Psi}{\partial y} = N = x \). Integrate \( M \) with respect to \( x \) to find \( \Psi(x,y) \) up to a function of \( y \).

Differentiate the expression for \( \Psi(x,y) \) with respect to \( y \) and set it equal to \( N = x \) to determine the function of \( y \) introduced during integration.

Use the initial condition \( y(\pi/2) = 0 \) to find the constant of integration in the implicit solution \( \Psi(x,y) = C \), and write the final implicit solution to the initial value problem.

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주요 개념

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

Initial Value Problem (IVP)

An initial value problem involves solving a differential equation with a given condition specifying the value of the unknown function at a particular point. This condition helps determine the unique solution that fits the problem's context.

Exact Differential Equations

An exact differential equation can be written in the form M(x,y)dx + N(x,y)dy = 0, where there exists a function F(x,y) such that dF = M dx + N dy. Recognizing exactness allows solving by finding this potential function F.

Integrating Factor

When a differential equation is not exact, an integrating factor is a function used to multiply the equation to make it exact. Finding the correct integrating factor simplifies solving the equation by enabling the use of exact equation methods.

In Exercises 23–28, solve the initial value problem.x dy + (y - cos x) dx = 0, y(π/2) = 0