11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(x) = x/y, y(2) = 4

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Identify the type of differential equation given: \( y'(x) = \frac{x}{y} \). This is a separable differential equation because the right side can be expressed as a function of \( x \) divided by a function of \( y \).

Rewrite the differential equation by separating variables: multiply both sides by \( y \) and multiply both sides by \( dx \) to get \( y \, dy = x \, dx \).

Integrate both sides: integrate \( y \, dy \) with respect to \( y \) and \( x \, dx \) with respect to \( x \). This gives \( \int y \, dy = \int x \, dx \).

After integrating, you will have an implicit solution involving \( y \) and \( x \) plus a constant of integration \( C \). Use the initial condition \( y(2) = 4 \) to solve for \( C \).

Finally, write the explicit solution for \( y \) in terms of \( x \) by solving the implicit equation for \( y \), if possible.

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주요 개념

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Separable Differential Equations

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

Initial Value Problems (IVP)

An initial value problem specifies a differential equation along with a condition that the solution must satisfy at a particular point. This condition helps determine the unique solution by solving for the constant of integration after finding the general solution.

Integration Techniques

Solving separable equations requires integrating both sides after separating variables. Familiarity with basic integration rules and techniques, such as power rule and substitution, is essential to find the explicit form of the solution.

11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.y′(x) = x/y, y(2) = 4