Use the integrating factor method to find the general solution of the linear first order ODE x\frac{dy}{dx} - y =x^2 e^x .The general solution of the ODE can be written as

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The given ODE is x dy/dx - y = x^2 e^x. To use the integrating factor method, first put it in standard linear form: dy/dx - (1/x) y = x e^x.

Next, compute the integrating factor μ(x) = exp(∫ -1/x dx) = exp(-ln|x|) = 1/|x|. For convenience and to keep things simple on an interval where x>0 (the same result holds on x<0 with appropriately absorbed constants), we take μ(x) = 1/x. Multiply every term by μ: (1/x) dy/dx - (1/x^2) y = e^x. The left-hand side is the derivative of y/x, since d/dx (y/x) = (x dy/dx - y)/x^2, which, when multiplied by x^2 and then divided by x^2, aligns with the left-hand side after simplifying, giving d/dx (y/x) = e^x.

Integrate both sides with respect to x: y/x = ∫ e^x dx = e^x + C. Therefore y = x e^x + C x.

Thus the general solution is y(x) = x e^x + C x, which matches the provided candidate form. Since no alternative answer options were supplied in the prompt, we cannot compare choices, but note that any algebraically equivalent expression (e.g., y = x(e^x + C)) represents the same family of solutions because the constant can be reabsorbed into the product with x. Also mindful, the integrating factor method requires x ≠ 0, so the solution is valid on intervals that do not cross x=0.

PHAS0009_24-25 4.3 Quiz 4 [0.25 hrs]

Use the integrating factor method to find the general solution of the linear first order ODE x\frac{dy}{dx} - y =x^2 e^x .The general solution of the ODE can be written as

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