Word Count : 518

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We are asked to solve the differential equation manually: $$ \frac{dy}{dx} + xy = y^2 e^{x^2/2} \sin x $$ Let's solve it step by step. --- ### Step 1: Recognize the type The given equation is: $$ \frac{dy}{dx} + P(x)y = Q(x)y^n $$ with: * $P(x) = x$ * $Q(x) = e^{x^2/2} \sin x$ * $n = 2$ This is a Bernoulli differential equation of the form: $$ \frac{dy}{dx} + P(x)y = Q(x)y^n $$ --- ### Step 2: Apply Bernoulli substitution For a Bernoulli equation: $$ \frac{dy}{dx} + P(x)y = Q(x)y^n $$ Use substitution: $$ v = y^{1-n} \implies v = y^{1-2} = _________ _______ _______ _______ ______ __________ _________ _________ _____ __________ ________ ______.
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