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

We are given the differential equation: \[ (1 + x^2) \frac{dy}{dx} + 2xy - 4x^2 = 0. \] Rearrange the equation: \[ (1 + x^2) \frac{dy}{dx} = 4x^2 - 2xy. \] Now, divide both sides by \( 1 + x^2 \) to isolate \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{4x^2 - 2xy}{1 + x^2}. \] This is a first-order linear differential equation. We can solve it using an appropriate method such as an integrating factor or substitution. However, a more direct approach involves solving for the particular solution using the initial condition \( y(0) = 0 \). Substituting \(x = 0\) into the equation: \[ \frac{dy}{dx} = \frac{4(0)^2 - 2(0)y}{1 + (0)^2} = 0. \] Thus, \( y(0) = 0 \) satisfies the initial condition, and the general solution is: \[ y(x) = \text{constant} \quad \Rightarrow \quad y(x) = 0 \text{ (since the initial condition is 0)}. \] Thus, the solution to the differential equation is \( y(x) = 0 \).