Question

Solve the differential equation \((x^2 + y^2) \, dx + xy \, dy = 0\), \(y(1) = 1\).

Hint:

When solving nonlinear differential equations like this one, it's often helpful to use substitution methods such as \(v = \frac{y}{x}\). This can simplify the equation and make it easier to solve. Don't forget to check the boundary conditions to find the specific solution that fits the given problem.

Answer 1

The differential equation \((x^2 + y^2) \, dx + xy \, dy = 0\) is addressed. The variables are separated by rewriting the equation as: \[ \frac{dy}{dx} = -\frac{x^2 + y^2}{xy} \] This is a nonlinear first-order differential equation. A substitution method is employed: let \(v = \frac{y}{x}\), implying \(y = vx\) and \(dy = v \, dx + x \, dv\). Substituting \(y = vx\) and \(dy = v \, dx + x \, dv\) into the original equation and simplifying yields the solution for \(v\), and subsequently for \(y(x)\). The resulting solution is: \[ y(x) = \frac{1}{\sqrt{1 - x^2}} \] The initial condition \(y(1) = 1\) is applied to determine the solution under the given boundary conditions. Correct Answer:
The solution to the differential equation is: \[ y(x) = \frac{1}{\sqrt{1 - x^2}} \]