The given ordinary differential equation is:

$\frac{dQ}{dt} = -11Q$

This is a first-order linear differential equation, and it can be solved by separating the variables. Here's the step-by-step solution:

Step 1: Separate the variables

We want to rearrange the equation so that all terms involving $Q$ are on one side, and all terms involving $t$ are on the other side:

$\frac{1}{Q} \, dQ = -11 \, dt$

Step 2: Integrate both sides

Now integrate both sides:

$\int \frac{1}{Q} \, dQ = \int -11 \, dt$

The integrals are:

$\ln|Q| = -11t + C$

where $C$ is the constant of integration.

Step 3: Solve for $Q$

To solve for $Q$, we exponentiate both sides:

$|Q| = e^{(-11t + C)} = e^{-11t} e^C$

Since $e^C$ is just a constant, we can rewrite it as a new constant $A = e^C$:

$|Q| = A e^{-11t}$

Thus, the general solution is:

$Q(t) = A e^{-11t}$

where $A$ is a constant determined by initial conditions.

---

Would you like further details on the steps, or do you have any questions?

Related Questions:

1. What are the initial conditions needed to determine the constant $A$?
2. How does the solution behave as $t$ increases or decreases?
3. What is the physical interpretation of the constant $A$ in an applied context?
4. How would the solution change if the coefficient $-11$ were replaced by a different value?
5. Can you solve similar differential equations with non-constant coefficients?

Tip:

For first-order linear differential equations, separation of variables is a powerful and simple technique when the equation is separable.