Question

The solution of \[ \frac{dv}{dt} + \frac{k}{m} v = -g \] is:

• A. \( v = c e^{-\frac{k}{m} t} + \frac{mg}{k} \)
• B. \( v = c e^{\frac{k}{m} t} - \frac{mg}{k} \)
• C. \( v = c e^{\frac{k}{m} t} + \frac{mg}{k} \)
• D. \( v = c e^{-\frac{k}{m} t} - \frac{mg}{k} \)

Hint:

For first-order linear differential equations, use the integrating factor method to find the solution.

Answer 1

The Correct Option is A

Step 1: This is a first-order linear differential equation. The solution to such an equation is: \[ v(t) = c e^{-\frac{k}{m} t} + \frac{mg}{k}. \] Step 2: The constant \( c \) is determined by initial conditions.

Final Answer: \[ \boxed{c e^{-\frac{k}{m} t} + \frac{mg}{k}} \]