Solve the following differential equation with given initial condition.
y prime equals 11 ty minus 12 t​, y left parenthesis 0 right parenthesis equals 1
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Part 1
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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.

Solve the following differential equation with given initial condition: y' = 11ty - 12t, y(0) = 1

Learn how to solve the first-order linear differential equation y' = 11ty - 12t with the initial condition y(0) = 1. This solution walks through the steps of using the integrating factor method to find the exact solution. Ideal for undergraduate students in Calculus I or II.

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