Question

Solve: \( (1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0 \).

Hint:

The most crucial step in solving linear differential equations is to correctly identify \( P(x) \) and \( Q(x) \) by converting the equation to its standard form. A common error is failing to divide the entire equation by the coefficient of \( \frac{dy}{dx} \). Always ensure the coefficient of \( \frac{dy}{dx} \) is 1 before proceeding.

Answer 1

Step 1: Understanding the Concept:
This is a first-order linear differential equation. It can be solved by first arranging it into the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \) and then using the method of integrating factors.
Step 2: Key Formula or Approach:
1. Rewrite the equation in the standard form \( \frac{dy}{dx} + P(x)y = Q(x) \).
2. Calculate the Integrating Factor (I.F.) using the formula \( \text{I.F.} = e^{\int P(x)dx} \).
3. The solution is given by the formula \( y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) dx + C \).
Step 3: Detailed Explanation:
The given equation is: \[ (1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0 \] First, rearrange it to isolate the term with \( \frac{dy}{dx} \) and \( y \): \[ (1 + x^2)\frac{dy}{dx} + 2xy = 4x^2 \] Now, divide by \( (1 + x^2) \) to get the standard form: \[ \frac{dy}{dx} + \frac{2x}{1 + x^2}y = \frac{4x^2}{1 + x^2} \] Comparing this with \( \frac{dy}{dx} + P(x)y = Q(x) \), we have: \[ P(x) = \frac{2x}{1 + x^2} \quad \text{and} \quad Q(x) = \frac{4x^2}{1 + x^2} \] Next, we calculate the integrating factor: \[ \text{I.F.} = e^{\int P(x)dx} = e^{\int \frac{2x}{1 + x^2}dx} \] Let \( u = 1 + x^2 \), so \( du = 2x dx \). The integral becomes \( \int \frac{1}{u}du = \ln|u| = \ln(1 + x^2) \). \[ \text{I.F.} = e^{\ln(1 + x^2)} = 1 + x^2 \] Now, we use the solution formula: \[ y \cdot (1 + x^2) = \int \frac{4x^2}{1 + x^2} \cdot (1 + x^2) dx + C \] \[ y \cdot (1 + x^2) = \int 4x^2 dx + C \] Evaluate the integral: \[ y \cdot (1 + x^2) = 4 \left( \frac{x^3}{3} \right) + C \] \[ y(1 + x^2) = \frac{4}{3}x^3 + C \] Step 4: Final Answer:
The general solution to the differential equation is \( y(1 + x^2) = \frac{4}{3}x^3 + C \).