Discussion
The general solution of the differential equation \(\frac{dy}{dx} = e^{3x-2y} + x^{2}e^{-2y}\) is
Answer: A
The solution is \(C = \frac{1}{2}e^{2y} - \frac{1}{3}(e^{3x} + x^{3})\).
Factor out \(e^{-2y}\) and separate the variables:
\[ \begin{aligned} \frac{dy}{dx} &= e^{-2y}(e^{3x} + x^{2}) \\ e^{2y}\,dy &= (e^{3x} + x^{2})\,dx \\ \frac{1}{2}e^{2y} &= \frac{1}{3}e^{3x} + \frac{1}{3}x^{3} + C \end{aligned} \]
Rearranged, \(C = \frac{1}{2}e^{2y} - \frac{1}{3}(e^{3x} + x^{3})\). The options with \(x^{2}\) inside the bracket forget to integrate \(x^{2}\). The last option drops the \(\frac{1}{2}\) from integrating \(e^{2y}\); doubling everything would give \(e^{2y} - \frac{2}{3}(\ldots)\), not \(\frac{1}{3}\).
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