5.) Let vec F (x,y,z)= langle-3x,y^ 2 ,5 rangle (Gi) (2 pts) Compute the flow line vec r (t) for subject to the condition vec F; vec r (1)= langle2,-1,3 rangle .

To solve this problem, we need to find the flow line (or integral curve) of the vector field \(\vec{F}(x, y, z) = \langle -3x, y^2, 5 \rangle\) that passes through the point \(\vec{r}(1) = \langle 2, -1, 3 \rangle\).

### Step 1: Set Up the System of Differential Equations

The flow line \(\vec{r}(t) = \langle x(t), y(t), z(t) \rangle\) is found by solving the system of differential equations given by:

\[
\frac{dx}{dt} = -3x, \quad \frac{dy}{dt} = y^2, \quad \frac{dz}{dt} = 5
\]

### Step 2: Solve Each Differential Equation Separately

**1. Solve \(\frac{dx}{dt} = -3x\)**

This is a separable differential equation:

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

Integrating both sides:

\[
\ln |x| = -3t + C_1
\]

Exponentiating both sides:

\[
x(t) = C_1 e^{-3t}
\]

**2. Solve \(\frac{dy}{dt} = y^2\)**

Rewrite as:

\[
\frac{dy}{y^2} = dt
\]

Integrate both sides:

\[
-\frac{1}{y} = t + C_2
\]

Solve for \(y\):

\[
y(t) = -\frac{1}{t + C_2}
\]

**3. Solve \(\frac{dz}{dt} = 5\)**

Integrate:

\[
z(t) = 5t + C_3
\]

### Step 3: Apply the Initial Condition \(\vec{r}(1) = \langle 2, -1, 3 \rangle\)

Plug \(t = 1\) into each equation:

1. For \(x(1) = 2\):

   \[
   2 = C_1 e^{-3 \cdot 1} \implies C_1 = 2e^3
   \]

   So, \(x(t) = 2e^3 e^{-3t} = 2e^{3(1-t)}\).

2. For \(y(1) = -1\):

   \[
   -1 = -\frac{1}{1 + C_2} \implies C_2 = 0
   \]

   So, \(y(t) = -\frac{1}{t}\).

3. For \(z(1) = 3\):

   \[
   3 = 5 \cdot 1 + C_3 \implies C_3 = -2
   \]

   So, \(z(t) = 5t - 2\).

### Step 4: Write the Flow Line \(\vec{r}(t)\)

The flow line is:

\[
\vec{r}(t) = \langle 2e^{3(1-t)}, -\frac{1}{t}, 5t - 2 \rangle.
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do we solve differential equations of different types (separable, linear, etc.)?
2. How can initial conditions affect the solution of a differential equation?
3. What are flow lines and how do they relate to vector fields?
4. How do you find the direction of flow lines from a given vector field?
5. How does the exponential function behave as \(t\) approaches infinity or zero?

**Tip:** When solving differential equations, always verify your solution by substituting it back into the original equation to ensure consistency.

5.) Let vec F (x,y,z)= langle-3x,y^2 ,5 rangle (Gi) (2 pts) Compute the flow line vec r (t) for subject to the condition vec F; vec r (1)= langle2,-1,3 rangle.

Learn how to compute the flow line of a vector field F(x, y, z) = ⟨-3x, y^2, 5⟩ with the given condition r(1) = ⟨2, -1, 3⟩. This step-by-step solution covers solving separable differential equations, applying initial conditions, and determining the integral curve for the flow line. Suitable for undergraduate mathematics students.

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