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Find the general explicit solution to 1/x dy/dx − 2y/x^2 = (x ^2) e^-x , where x > 0.

User DumTux
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Final answer:

The general explicit solution to the given differential equation is y = x^2(Ce^x - e^-x + x + 1), where C is the constant of integration determined by initial conditions.

Step-by-step explanation:

The solution to the given ordinary differential equation can be obtained using the method of integrating factors. Since the equation is of the form 1/x dy/dx - 2y/x2 = x2e-x, the integrating factor, μ(x), is e∫( -2/x)dx which simplifies to 1/x2. Multiplying both sides of the equation by the integrating factor, we obtain (y/x2)' = e-x, where the prime denotes the derivative concerning x. Integrating both sides concerning x will give us the general solution. Integrate the right side using the power rule for integrating: ∫2y/x + x^4e^-x dx = ∫2/y dy + ∫x^4e^-x dx. The first integral gives us 2ln|y| + C1, and the second integral can be evaluated using integration by parts.

To find the solution, integrate both sides to get y/x2 = ∫ e-x dx + C, where C is the constant of integration. The integral of the right side can be solved by integration by parts, ultimately yielding y = x2(Cex - e-x + x + 1), where C is the arbitrary constant obtained after integration.

User Robin McCorkell
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