Question

Solve the differential equation \( (1 + x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0 \) subject to initial condition \( y(0) = 0 \).

Hint:

When solving first-order linear differential equations, use an appropriate method such as the integrating factor or substitution. Always apply the initial condition to find the particular solution.

Answer 1

The provided differential equation is: \[ (1 + x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0. \] Rearranging the equation yields: \[ (1 + x^2) \frac{dy}{dx} = 4x^2 - 2xy. \] Dividing both sides by \( 1 + x^2 \) to isolate \(\frac{dy}{dx}\) gives: \[ \frac{dy}{dx} = \frac{4x^2 - 2xy}{1 + x^2}. \] This is a first-order linear differential equation. While it can be solved using methods like an integrating factor or substitution, a direct approach using the initial condition \( y(0) = 0 \) is also possible. Substituting \(x = 0\) into the derived equation: \[ \frac{dy}{dx} = \frac{4(0)^2 - 2(0)y}{1 + (0)^2} = 0. \] This confirms that \( y(0) = 0 \) satisfies the initial condition. The general solution can be expressed as: \[ y(x) = \text{constant}. \] Given the initial condition \( y(0) = 0 \), the constant must be 0, leading to the particular solution: \[ y(x) = 0. \] Therefore, the solution to the differential equation under the given initial condition is \( y(x) = 0 \).