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The given differential equation is:

\[
\frac{dx}{dt} = 2(7t - 1)e^{-t} - 7xt
\]

This is a first-order linear non-homogeneous differential equation. To solve it, we can use the integrating factor method. Here's a step-by-step outline for solving this equation:

### Step 1: Rewrite the equation in standard form
The standard form of a linear differential equation is:

\[
\frac{dx}{dt} + P(t)x = Q(t)
\]

For this equation:
\[
P(t) = 7t, \quad Q(t) = 2(7t - 1)e^{-t}
\]

Thus, the equation becomes:

\[
\frac{dx}{dt} + 7tx = 2(7t - 1)e^{-t}
\]

### Step 2: Find the integrating factor
The integrating factor (IF) is given by:

\[
\mu(t) = e^{\int P(t) \, dt} = e^{\int 7t \, dt} = e^{\frac{7t^2}{2}}
\]

### Step 3: Multiply through by the integrating factor
Multiply both sides of the differential equation by the integrating factor \( \mu(t) = e^{\frac{7t^2}{2}} \):

\[
e^{\frac{7t^2}{2}} \frac{dx}{dt} + 7t e^{\frac{7t^2}{2}} x = 2(7t - 1)e^{-t} e^{\frac{7t^2}{2}}
\]

This simplifies the left side to:

\[
\frac{d}{dt} \left( x e^{\frac{7t^2}{2}} \right) = 2(7t - 1)e^{-t} e^{\frac{7t^2}{2}}
\]

### Step 4: Integrate both sides
Now, integrate both sides with respect to \( t \):

\[
x e^{\frac{7t^2}{2}} = \int 2(7t - 1)e^{-t} e^{\frac{7t^2}{2}} \, dt
\]

The right-hand side involves integrating a more complex function, which may require additional methods like integration by parts or approximations, depending on the specific form.

---

Would you like to proceed with the integration of the right-hand side or need further clarification on any step so far?

Here are 5 related questions that you might find interesting:
1. What is the general method for solving linear first-order differential equations?
2. How does the integrating factor method work in more complex scenarios?
3. Could this equation be solved using numerical methods?
4. How does the solution behavior change if \( P(t) \) and \( Q(t) \) have different forms?
5. Can this type of equation appear in real-world applications, such as physics or biology?

**Tip**: Always check whether a given differential equation is homogeneous or non-homogeneous, as this affects the solution method.

Solve the differential equation: dx/dt = 2(7t - 1)e^{-t} - 7xt

This solution covers the method of solving the first-order linear differential equation dx/dt = 2(7t - 1)e^{-t} - 7xt using the integrating factor technique. Detailed step-by-step explanation provided for Grade 11-12 or university students.

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