Solve the differential equation. $$\frac { d y } { d x } = - y - 8$$
A) $y = C e ^ { - x } + 8$
B) $y = C e ^ { x } - 8$
C) $y = C e ^ { x } + 8$
D) $y = C e ^ { - x } - 8$
E) $y = C e ^ { x } + 16$

Question

Quizplus · Accepted Answer

We begin by separating the variables and integrating:

$$\frac{dy}{dx}=-y-8$$
$$\frac{dy}{y+8}=-dx$$
$$\int\frac{1}{y+8}dy=\int-1dx$$
$$\ln|y+8|=-x+C$$
$$|y+8|=e^{-x+C}=e^{-x}e^{C}$$
Since $e^C$ is just a constant, we can replace it with a new constant $k$.
$$|y+8|=ke^{-x}$$
Let's now solve for $y$:

$$y + 8 = ke^{-x} \text{ or } y+8 = -ke^{-x}$$

Solving for $y$ in each case, we get:

$$y = ke^{-x} - 8 \text{ or } y = -ke^{-x}-8$$

Since $k$ is an arbitrary constant, we can just simplify and write:

$$y = Ce^{-x} - 8, \text{ where } C = -k$$

Thus, our answer is (D).

Solve the Differential Equation dydx=−y−8\frac { d y } { d x } = - y - 8dxdy​=−y−8

Solve the Differential Equation $\frac { d y } { d x } = - y - 8$