Solve the initial value problems in Exercises 115–120.
115. dy/dx = 1/√(1 - x²), y(0) = 0

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Identify the differential equation given: \(\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}\) with the initial condition \(y(0) = 0\).

Recognize that to find \(y\), you need to integrate the right-hand side with respect to \(x\): \(y = \int \frac{1}{\sqrt{1 - x^2}} \, dx + C\).

Recall the integral formula: \(\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin(x) + C\).

Apply the initial condition \(y(0) = 0\) to solve for the constant of integration \(C\): substitute \(x=0\) and \(y=0\) into \(y = \arcsin(x) + C\).

Write the final solution as \(y = \arcsin(x) + C\) with the value of \(C\) found from the initial condition.

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주요 개념

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

Separable Differential Equations

A separable differential equation can be written as a product of a function of x and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.

Integration of Inverse Trigonometric Functions

The integral of 1/√(1 - x²) is the inverse sine function, arcsin(x), plus a constant. Recognizing this form is essential for solving differential equations involving derivatives that match inverse trigonometric derivatives.

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a particular point, allowing determination of the constant of integration. This ensures a unique solution that satisfies both the differential equation and the initial condition.

Solve the initial value problems in Exercises 115–120.115. dy/dx = 1/√(1 - x²), y(0) = 0