The general solution of the differential equation \( \displaystyle\frac{dy}{dx}=e^{2x-y}+xe^{-y} \) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 2e^{y}=e^{2x}+x^2+C \): where C is an arbitrary constant
Taking \( e^{-y} \) common on the right, the variables separate:
\( \displaystyle \frac{dy}{dx}=e^{-y}\left(e^{2x}+x\right) \)
Multiplying both sides by \( e^{y} \):
\( \displaystyle e^{y}\,dy=\left(e^{2x}+x\right)dx \)
Integrating:
\( \displaystyle e^{y}=\frac{e^{2x}}{2}+\frac{x^2}{2}+C_1 \)
Multiplying by 2 and writing C for \( 2C_1 \):
\( \displaystyle 2e^{y}=e^{2x}+x^2+C \)
Why the other options are incorrect
Option (1) \( 2e^{-y}=e^{2x}+x+C \): where C is an arbitrary constant — Incorrect. Separating the variables puts \( e^{y} \) (not \( e^{-y} \)) with dy, and \( \int x\,dx=\frac{x^2}{2} \) gives \( x^2 \), not x, after multiplying by 2.
Option (3) \( 2e^{-y}=e^{2x}+x^2+C \): where C is an arbitrary constant — Incorrect. The left side should be \( 2e^{y} \): dividing by \( e^{-y} \) gives \( e^{y}\,dy \).
Option (4) \( 2e^{y}=e^{2x}+x+C \): where C is an arbitrary constant — Incorrect. The integral of x is \( \frac{x^2}{2} \), so the term is \( x^2 \), not x.