We are given the first-order linear differential equation:

$\frac{dy}{dt} = 11ty - 12t$

with the initial condition $y(0) = 1$.

Step 1: Rearrange the equation

We start by writing the differential equation in the form: $\frac{dy}{dt} - 11ty = -12t$

This is a first-order linear differential equation, and we can solve it using an integrating factor.

Step 2: Find the integrating factor

The integrating factor $\mu(t)$ is given by:

$\mu(t) = e^{\int -11t dt} = e^{-11t^2/2}$

Step 3: Multiply the equation by the integrating factor

Now, multiply the entire differential equation by the integrating factor $\mu(t) = e^{-11t^2/2}$:

$e^{-11t^2/2} \frac{dy}{dt} - 11t e^{-11t^2/2} y = -12t e^{-11t^2/2}$

The left-hand side is the derivative of $e^{-11t^2/2} y$, so we can rewrite the equation as:

$\frac{d}{dt} \left( e^{-11t^2/2} y \right) = -12t e^{-11t^2/2}$

Step 4: Integrate both sides

Now, integrate both sides with respect to $t$:

$e^{-11t^2/2} y = \int -12t e^{-11t^2/2} dt$

The integral on the right-hand side can be computed by recognizing it as a standard form:

$\int -12t e^{-11t^2/2} dt = e^{-11t^2/2} + C$

Step 5: Solve for $y(t)$

Now, solve for $y(t)$ by dividing both sides by $e^{-11t^2/2}$:

$y(t) = 1 + C e^{11t^2/2}$

Step 6: Use the initial condition

We use the initial condition $y(0) = 1$ to find $C$:

$1 = 1 + C e^0$

So, $C = 0$.

Final solution:

Thus, the solution to the differential equation is:

$y(t) = 1$

Would you like more details on any of these steps?

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Here are 5 related questions:

1. What are the general steps for solving first-order linear differential equations?
2. How do you compute the integrating factor for a first-order linear differential equation?
3. Why is the integrating factor method effective for solving differential equations?
4. Can you solve non-linear first-order differential equations with similar methods?
5. How do initial conditions help in finding the specific solution to a differential equation?

Tip: Always check your solution by substituting it back into the original differential equation to verify its correctness.