Question

Let \(y=f(x)\) satisfy \[ \frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^6+4x}{\sqrt{1-x^2}},\ -1<x<1 \] with \(f(0)=0\). If \[ 6\int_{-1/2}^{1/2} f(x)\,dx = 2\pi-\alpha, \] then \(\alpha^2\) equals:

• A. \(25\)
• B. \(26\)
• C. \(27\)
• D. \(28\)

Hint:

Always check symmetry before integrating on \([-a,a]\).

Answer 1

The Correct Option is C

Concept: Use integrating factor for linear differential equations.

Step 1: Find I.F. \[ P=\frac{x}{x^2-1} \] \[ I.F.=e^{\int \frac{x}{x^2-1}dx} =\sqrt{1-x^2} \]

Step 2: Convert to exact derivative. \[ \frac{d}{dx}(y\sqrt{1-x^2})=x^6+4x \]

Step 3: Integrate. \[ y\sqrt{1-x^2}=\frac{x^7}{7}+2x^2 \] \[ y=\frac{\frac{x^7}{7}+2x^2}{\sqrt{1-x^2}} \]

Step 4: Use symmetry on limits. \[ 6\int_{-1/2}^{1/2} f(x)\,dx = 2\pi-\alpha \Rightarrow \alpha^2=27 \] \[ \boxed{27} \]