Q87 / 180 · COMEDK 2025 (25 May Shift) · Mathematics
The solution of the differential equation $\dfrac{d y}{d x}=e^{x+y}+x^2 e^y$ is
- A$e^{x-y}+\dfrac{x^3}{3}=c$
- B$e^x-e^{-y}+\dfrac{x^3}{3}=c$
- Correct answerC$e^x+e^{-y}+\dfrac{x^3}{3}=c$
- D$e^x-e^{-y}=\dfrac{x^3}{3}+c$
Solution
The given differential equation is $\dfrac{dy}{dx} = e^{x+y} + x^2 e^y$.
Factoring out $e^y$ from the right side, we get $\dfrac{dy}{dx} = e^y(e^x + x^2)$.
Separating the variables, we have $\dfrac{dy}{e^y} = (e^x + x^2) dx$, which can be written as $e^{-y} dy = (e^x + x^2) dx$.
Integrating both sides, we get $\int e^{-y} dy = \int (e^x + x^2) dx$.
Performing the integration, we obtain $-e^{-y} = e^x + \dfrac{x^3}{3} + c$.
Rearranging the terms, we get $e^x + e^{-y} + \dfrac{x^3}{3} = -c$. Since $-c$ is an arbitrary constant, we can write this as $e^x + e^{-y} + \dfrac{x^3}{3} = C$.
Answer: $e^x+e^{-y}+\dfrac{x^3}{3}=c$
Factoring out $e^y$ from the right side, we get $\dfrac{dy}{dx} = e^y(e^x + x^2)$.
Separating the variables, we have $\dfrac{dy}{e^y} = (e^x + x^2) dx$, which can be written as $e^{-y} dy = (e^x + x^2) dx$.
Integrating both sides, we get $\int e^{-y} dy = \int (e^x + x^2) dx$.
Performing the integration, we obtain $-e^{-y} = e^x + \dfrac{x^3}{3} + c$.
Rearranging the terms, we get $e^x + e^{-y} + \dfrac{x^3}{3} = -c$. Since $-c$ is an arbitrary constant, we can write this as $e^x + e^{-y} + \dfrac{x^3}{3} = C$.
Answer: $e^x+e^{-y}+\dfrac{x^3}{3}=c$
Hence, the correct answer is option: 'C'.