Question

The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is ______.

Options

• xe^y + x² = C

• xe^y + y² = C

• ye^x + x² = C

• ye^y + x² = C

Answer 1

The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is ye^x + x² = C.

Explanation:

The given equation

e^xdy + (ye^x + 2x) dx = 0

or ${\displaystyle e^x\frac{dy}{dx}+y{e}^x+2x=0}$

${\displaystyle \frac{dy}{dx}+1\cdot y=\frac{-2x}{e^x}}$

Comparing this equation with ${\displaystyle \frac{dy}{dx}+Py=Q.}$

P = 1, Q = ${\displaystyle \frac{-2x}{e^x}}$

∴ ${\displaystyle I.F.=e^{\int 1dx}=e^x}$

Hence, the general solution of the equation

${\displaystyle y\cdot e^x=\int \frac{-2x}{e^x}\cdot e^{x}dx+C}$

${\displaystyle y{e}^x=\int -2xdx+C}$

${\displaystyle y{e}^x=-2\frac{x^2}{2}+C}$

${\displaystyle y{e}^x=-x^2+C}$

${\displaystyle y{e}^x+x^2=C}$