Related Formula
For a linear differential equation (dy)/(dx) + Py = Q$\frac{dy}{dx} + Py = Q$, the Integrating Factor (IF) is defined as:
IF = e∫ P dx$$\text{IF} = e^{\int P \, dx}$$
Core Logic
Divide the full differential equation by (x²+1)$(x^2+1)$:
(dy)/(dx) - ((2x)/(x²+1))y = ((x²+1)² x)/(x²+1) = (x²+1) x$$\frac{dy}{dx} - \left(\frac{2x}{x^2+1}\right)y = \frac{(x^2+1)^2 \cos x}{x^2+1} = (x^2+1)\cos x$$
This is a standard Linear Differential Equation with:
P = -(2x)/(x²+1), Q = (x²+1) x$$P = -\frac{2x}{x^2+1}, \quad Q = (x^2+1)\cos x$$
IF = e∫ -(2x)/(x²+1) dx = e-ln(x²+1) = (1)/(x²+1)$$\text{IF} = e^{\int -\frac{2x}{x^2+1}\,dx} = e^{-\ln(x^2+1)} = \frac{1}{x^2+1}$$
Step 1: Solve for General Solution
The solution format is y · IF = ∫ Q · IF dx$y \cdot \text{IF} = \int Q \cdot \text{IF} \, dx$:
y · (1)/(x²+1) = ∫ (x²+1) x · (1)/(x²+1) dx$$y \cdot \frac{1}{x^2+1} = \int (x^2+1)\cos x \cdot \frac{1}{x^2+1} \, dx$$
(y)/(x²+1) = x + c$$\frac{y}{x^2+1} = \sin x + c$$
Using the boundary condition y(0) = 1$y(0) = 1$:
(1)/(0+1) = (0) + c c = 1$$\frac{1}{0+1} = \sin(0) + c \implies c = 1$$
y = (x²+1)(sin x + 1)$$y = (x^2+1)(sin x + 1)$$
Step 2: Definite Integration Evaluation
We need to evaluate ∫₋₃³ y dx$\int_{-3}^{3} y \, dx$:
∫₋₃³ (x²+1)(sin x + 1) dx = ∫₋₃³ (x² x + x² + x + 1) dx$$\int_{-3}^{3} (x^2+1)(sin x + 1) \, dx = \int_{-3}^{3} (x^2\sin x + x^2 + \sin x + 1) \, dx$$
By symmetry of odd/even functions over symmetric intervals [-a, a]$[-a, a]$:
∫₋₃³ x² x dx = 0$\int_{-3}^{3} x^2\sin x \, dx = 0$ (since it is an odd function)
∫₋₃³ x dx = 0$\int_{-3}^{3} \sin x \, dx = 0$ (since it is an odd function)
Thus, we are left with the even components:
∫₋₃³ (x² + 1) dx = 2 ∫₀³ (x² + 1) dx = 2 [ (x³)/(3) + x ]₀³ = 2(9 + 3) = 24$$\int_{-3}^{3} (x^2 + 1) \, dx = 2 \int_{0}^{3} (x^2 + 1) \, dx = 2 \left[ \frac{x^3}{3} + x \right]_{0}^{3} = 2(9 + 3) = 24$$
Pattern Recognition
Splitting a symmetric interval integral into odd and even parts immediately simplifies calculations by dropping all odd functions down to zero.
Chapter Mix
Class 12 Mathematics: Differential Equations
Class 12 Mathematics: Integral Calculus
