Question

Solve: \( \frac{dy}{dx} = \frac{x^2 + y^2}{2xy} \).

Hint:

For differential equations involving separable variables, separate the variables, then integrate both sides to solve.

Answer 1

Step 1: Rearrange the equation to separate the variables \( y \) and \( x \): \[ \frac{dy}{dx} = \frac{x^2 + y^2}{2xy} \quad \Rightarrow \quad \frac{dy}{dx} = \frac{1}{2} \left( \frac{x}{y} + \frac{y}{x} \right) \] Step 2: Now, separate the variables: \[ \frac{dy}{\left( \frac{y}{x} + \frac{x}{y} \right)} = \frac{1}{2} dx \] Step 3: Integrate both sides and solve for \( y \) in terms of \( x \). Thus, the solution is obtained by performing the integration and solving the equation.