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We are given a velocity field:

\[
\vec{V} = \begin{pmatrix} 10y + t \\ 5x \end{pmatrix}
\]

### Part a) Streamline Equation

To find the equation of the streamline (línea de corriente), we know that streamlines satisfy:

\[
\frac{dy}{dx} = \frac{V_y}{V_x}
\]

From the velocity field, we have:

\[
V_x = 5x, \quad V_y = 10y + t
\]

Thus, the streamline equation is:

\[
\frac{dy}{dx} = \frac{10y + t}{5x}
\]

To solve this, we separate variables:

\[
\frac{dy}{10y + t} = \frac{dx}{5x}
\]

Now, we can integrate both sides. The integral becomes:

\[
\int \frac{1}{10y + t} dy = \int \frac{1}{5x} dx
\]

On the left-hand side, the integral is:

\[
\frac{1}{10} \ln |10y + t| = \frac{1}{5} \ln |x| + C
\]

Multiplying both sides by 10 to simplify:

\[
\ln |10y + t| = 2 \ln |x| + C'
\]

Exponentiating both sides:

\[
10y + t = A x^2
\]

where \(A = e^{C'}\) is a constant. Therefore, the equation of the streamline is:

\[
10y + t = A x^2
\]

where \(A\) is determined by the initial conditions \(x_0, y_0\).

---

### Part b) Particle Trajectory

For particle trajectory, we must solve the differential equations:

\[
\frac{dx}{dt} = V_x = 5x
\]
\[
\frac{dy}{dt} = V_y = 10y + t
\]

#### Solving for \(x(t)\):

\[
\frac{dx}{dt} = 5x
\]

This is a separable differential equation:

\[
\frac{dx}{x} = 5 dt
\]

Integrating both sides:

\[
\ln |x| = 5t + C_1
\]

Exponentiating:

\[
x = C_2 e^{5t}
\]

where \(C_2 = e^{C_1}\) is a constant determined by the initial condition \(x(0) = x_0\):

\[
x = x_0 e^{5t}
\]

#### Solving for \(y(t)\):

We have the equation:

\[
\frac{dy}{dt} = 10y + t
\]

This is a linear non-homogeneous differential equation. Using an integrating factor, \(\mu(t) = e^{-10t}\), we multiply both sides:

\[
e^{-10t} \frac{dy}{dt} = e^{-10t}(10y + t)
\]

The left-hand side becomes:

\[
\frac{d}{dt} (y e^{-10t}) = t e^{-10t}
\]

Integrating both sides:

\[
y e^{-10t} = -\frac{t}{10} e^{-10t} - \frac{1}{100} e^{-10t} + C_3
\]

Multiplying by \(e^{10t}\) to isolate \(y\):

\[
y = -\frac{t}{10} - \frac{1}{100} + C_3 e^{10t}
\]

Applying the initial condition \(y(0) = y_0\):

\[
y_0 = -\frac{0}{10} - \frac{1}{100} + C_3
\]
\[
C_3 = y_0 + \frac{1}{100}
\]

Thus, the equation for \(y(t)\) is:

\[
y(t) = -\frac{t}{10} - \frac{1}{100} + \left(y_0 + \frac{1}{100}\right) e^{10t}
\]

---

### Part c) Feasibility of Plant Locations

We need to determine if either of the given locations for the plant will cause smoke to reach the town \((5, 5)\) by time \(t = 10\).

- First location: \((-320, -120)\)
- Second location: \((-130, -45)\)

The goal is to check if particles starting from these locations at \(t = 0\) will reach \((5, 5)\) at \(t = 10\).

For both locations, we can use the particle trajectory equations \(x(t)\) and \(y(t)\) derived in part b.

Let me compute the specifics for both locations.For \( t = 10 \), the final positions of the particles starting from the two locations are:

1. **For the first location \((-320, -120)\):**
   - Final \(x\)-position: \( -320 \cdot e^{50} \)
   - Final \(y\)-position: \( -119.99 \cdot e^{100} - 1.01 \)

2. **For the second location \((-130, -45)\):**
   - Final \(x\)-position: \( -130 \cdot e^{50} \)
   - Final \(y\)-position: \( -44.99 \cdot e^{100} - 1.01 \)

Since \( e^{50} \) and \( e^{100} \) are extremely large numbers, both positions will be very far from the target point \((5, 5)\). Therefore, **neither location** will cause the smoke to reach the town by \( t = 10 \).

---

Let me know if you need more details or explanations, and here are some related questions to further explore:

1. How can we generalize this solution to any arbitrary velocity field?
2. What are the physical interpretations of streamlines and particle trajectories in fluid mechanics?
3. How does the behavior of exponential functions affect long-term predictions in dynamical systems?
4. Can we apply this method to analyze 3D flow fields?
5. What would change in the solution if we had variable coefficients in the velocity field?

**Tip:** When working with differential equations, always verify the boundary conditions or initial values to ensure the solution applies correctly to the specific case at hand.

Conocemos el campo de velocidad dado por: \( \vec{V} = \begin{pmatrix} 10y + t \\ 5x \end{pmatrix} \). Determine la ecuación de la línea de corriente y la trayectoria de una partícula que pasa por una posición arbitraria, y evalúe la viabilidad de dos ubicaciones para la planta de celulosa.

Learn how to solve for the streamlines and particle trajectories in a velocity field \( \vec{V} = \begin{pmatrix} 10y + t \\ 5x \end{pmatrix} \), including differential equations for streamlines and analyzing the viability of plant locations.

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