Question

Find the general solution of the differential equation y^2/2+2ye^t+(y+e^t)dy/dt = 0

Answer 1

The differential equation can be rewritten as follows:

$$\frac12 y^2 + 2 y e^t + \left( y + e^t \right) \frac{dy}{dt} \\ {} = \frac12 y^2 + y e^t + \frac{d}{dt} \left( \frac12 y^2 + y e^t \right) = 0 \, .$$

Denoting $u = \frac12 y^2 + y e^t$, we have the equation $u + du/dt = 0$, the general solution of which is $u = C_1 e^{-t}$, where $C_1$ is the integration constant. Hence, we have $\frac12 y^2 + y e^t = C_1 e^{-t}$, or $y^2 + 2 y e^t - C e^{-t} = 0$, where $C = 2 C_1$ is another integration constant. Solving this quadratic equation with respect to y, we obtain the final answer:

$$y = - e^t \mp \sqrt{ e^{2t} + C e^{-t}} = - e^t \left( 1 \pm \sqrt{1 + C e^{- 3 t}} \right) \, .$$