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We are tasked with finding the general solution of the equation: \[ x\frac{dy}{dx} + 4y = x^5 e^x \] using the method of variation of parameters. ### Step 1: Solve the homogeneous equation The homogeneous part of the equation is: \[ x \frac{dy}{dx} + 4y = 0 \] This is a linear first-order equation, which can be written as: \[ \frac{dy}{dx} + \frac{4}{x} y = 0 \] To solve this, we first find the integrating factor. The integrating factor \( \mu(x) \) is given by: \[ \mu(x) = e^{\int \frac{4}{x} dx} = e^{4 \ln x} = x^4 \] Multiplying both sides of the differential equation by the integrating factor: \[ x^4 \frac{dy}{dx} + 4x^3 y = 0 \] This simplifies to: \[ \frac{d}{dx}(x^4 y) = 0 \] Integrating both sides: \[ x^4 y ___ ____ ______ _____ ___ _________ _____.
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