The given equation is a first-order differential equation:

\[
\frac{dx}{dt} = 0.12t - \frac{3x}{50}
\]

### Steps to solve the differential equation:

1. **Separate the variables:**
   Rearrange the equation to isolate terms involving \(x\) and \(t\) on opposite sides of the equation.

   \[
   \frac{dx}{dt} + \frac{3x}{50} = 0.12t
   \]

   Now, this equation is in standard linear form: \(\frac{dx}{dt} + p(t) x = q(t)\), where \(p(t) = \frac{3}{50}\) and \(q(t) = 0.12t\).

2. **Find the integrating factor:**
   The integrating factor is given by \( \mu(t) = e^{\int p(t) dt} = e^{\int \frac{3}{50} dt} = e^{\frac{3}{50} t} \).

3. **Multiply both sides of the differential equation by the integrating factor:**

   \[
   e^{\frac{3}{50} t} \frac{dx}{dt} + e^{\frac{3}{50} t} \cdot \frac{3}{50} x = e^{\frac{3}{50} t} \cdot 0.12t
   \]

   The left-hand side becomes the derivative of \(x \cdot e^{\frac{3}{50} t}\), so the equation simplifies to:

   \[
   \frac{d}{dt} \left( x \cdot e^{\frac{3}{50} t} \right) = e^{\frac{3}{50} t} \cdot 0.12t
   \]

4. **Integrate both sides:**

   Integrating the left-hand side gives \(x \cdot e^{\frac{3}{50} t}\). Now, integrate the right-hand side:

   \[
   x \cdot e^{\frac{3}{50} t} = \int e^{\frac{3}{50} t} \cdot 0.12t \, dt
   \]

   To solve this integral, you can apply integration by parts.

5. **Solve for \(x(t)\):**

   After performing the integration and isolating \(x(t)\), you can then solve for \(x\).

Would you like me to complete the integration and provide the exact solution for \(x(t)\)? 

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Here are some questions to extend your understanding:
1. What are integrating factors and how are they derived?
2. How do you perform integration by parts?
3. Can you apply this method to non-linear differential equations?
4. What is the significance of initial conditions in solving differential equations?
5. How does the integrating factor method differ from other solution methods?

**Tip:** Always check for linearity before applying integrating factors, as it's specific to linear differential equations.

Explore the detailed solution for the first-order linear differential equation dx/dt = 0.12t - 3x/50. This problem involves using integrating factors and integration by parts to find the general solution. Suitable for students at a college level, particularly in Calculus II or Differential Equations courses.

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