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

We are given the differential equation \((x^2 + y^2) \, dx + xy \, dy = 0\). To solve it, we first separate the variables. Rewriting the equation: \[ \frac{dy}{dx} = -\frac{x^2 + y^2}{xy} \] This is a nonlinear first-order differential equation. To solve it, we try using a substitution method. Let \(v = \frac{y}{x}\), so that \(y = vx\), and therefore, \(dy = v \, dx + x \, dv\). Substitute \(y = vx\) and \(dy = v \, dx + x \, dv\) into the original equation. Substitute these expressions and simplify the equation to get the solution for \(v\), and ultimately for \(y(x)\). After performing the calculations, we get the solution as: \[ y(x) = \frac{1}{\sqrt{1 - x^2}} \] Finally, apply the initial condition \(y(1) = 1\) to obtain the solution for the given boundary conditions. Correct Answer:
The solution to the differential equation is: \[ y(x) = \frac{1}{\sqrt{1 - x^2}} \]