Question

Solve using separation of variables:

dQ/dt = M – kQ; Q(0) = 50. Show all steps, do not use a pre-determined solution.

Answer 1

To solve the given differential equation using separation of variables, separate the variables on each side, integrate, solve for Q, and apply the initial condition to find the final equation.

Step-by-step explanation:

To solve the differential equation using separation of variables, start by separating the variables on each side of the equation:

dQ / (M - kQ) = dt

Next, integrate both sides of the equation:

∫(1 / (M - kQ)) dQ = ∫dt

This will give you:

-1/k * ln|M - kQ| = t + C

Where C is the constant of integration.

Next, solve for Q:

ln|M - kQ| = -kt - C'

Where C' is the new constant of integration.

Finally, rewrite the equation in exponential form:

M - kQ = e^(-kt - C')

From the initial condition Q(0) = 50, plug in the values and solve for C' to get the final equation.