Final answer:
To solve the differential equation xy′ − 2y = x³ cos x, we use the method of integrating factors. The solution is y = (1/2x)(x⁴ sin x - 2xy²) + 5π²/(4x).
Step-by-step explanation:
To solve the differential equation xy′ − 2y = x³ cos x, we will use the method of integrating factors. First, we rearrange the equation to isolate the derivative of the unknown function y: y′ = (x³ cos x - 2y)/x. The integrating factor is then found by multiplying both sides of the equation by x: xy′ = x²³ cos x - 2xy. The left side of the equation is now the derivative of the product xy with respect to x, so the equation becomes d/dx(xy) = x²³ cos x - 2xy.
Integrating both sides of the equation gives us xy = ∫(x²³ cos x - 2xy) dx. Simplifying and evaluating the integral, we have xy = (1/24)x⁴ sin x - x²y²/2 + C, where C is the constant of integration. Finally, solving for y, we have y = (1/2x)(x⁴ sin x - 2xy²) + C/x.
To find the particular solution that satisfies the initial condition y(π) = π², we substitute x = π and y = π² into the equation. We have π² = (1/2π)(π⁴ sin π - 2π(π²)²) + C/π. Simplifying further, we find C = 5π²/4. Therefore, the solution to the given differential equation is y = (1/2x)(x⁴ sin x - 2xy²) + 5π²/(4x).