The general solution of the differential equa | vedclass

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Question

(A) Given the differential equation: $\frac{dy}{dx} = e^{x+y} + x^2 e^{x^3+y}$ Separating the terms: $\frac{dy}{dx} = e^y(e^x + x^2 e^{x^3})$ Rearrang

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$e^{-y} + e^x + \frac{1}{3} e^{x^3} = C$

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(A) Given the differential equation: $\frac{dy}{dx} = e^{x+y} + x^2 e^{x^3+y}$ Separating the terms: $\frac{dy}{dx} = e^y(e^x + x^2 e^{x^3})$ Rearranging the variables: $e^{-y} dy = (e^x + x^2 e^{x^3}) dx$ Integrating both sides: $\int e^{-y} dy = \int (e^x + x^2 e^{x^3}) dx$ Solving the integrals: $-e^{-y} = e^x + \frac{1}{3} e^{x^3} + C$ Rearranging to standard form: $e^{-y} + e^x + \frac{1}{3} e^{x^3} = -C$ Since $-C$ is also a constant,we can write: $e^{-y} + e^x + \frac{1}{3} e^{x^3} = C$

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y} + x^2 e^{x^3+y}$ is (where $C$ is a constant of integration):

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