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 `e^x dy/dx + ye^x + 2x = 0`

`dy/dx + 1 * y = (- 2x)/e^x`

Comparing this equation with `dy/dx + Py = Q.`

P = 1, Q = `(- 2x)/e^x`

∴ `I.F. = e^(int 1 dx) = e^x`

Hence, the general solution of the equation

`y * e^x = int (- 2x)/e^x * e^x dx + C`

`y e^x = int - 2 x dx + C`

`y e^x = - 2 x^2/2 + C`

`y e^x = - x^2 + C`

`y e^x + x^2 = C`