6. Given that $$ \frac{d y}{d x}=2 x+7 $$ and $$ y=5 $$ when $$ x=-1 $$, find the value of $$ y $$ when $$ x=2 $$ 7. Given that $$ \frac{d v}{d t}=3-2 t $$ and $$ v=2 $$ when $$ t=1 $$, find the value of $$ v $$ when $$ t=2 $$

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### **Problem 6**

**Given:**
- $$\frac{dy}{dx} = 2x + 7$$
- $$y = 5$$ when $$x = -1$$

**Find:**
- The value of $$y$$ when $$x = 2$$.

**Solution:**

1. **Integrate $$\frac{dy}{dx}$$ to find $$y$$:**
   $$
   \frac{dy}{dx} = 2x + 7 \implies y = \int (2x + 7) \, dx = x^2 + 7x + C
   $$
   where $$C$$ is the constant of integration.

2. **Apply the initial condition $$y = 5$$ when $$x = -1$$ to find $$C$$:**
   $$
   5 = (-1)^2 + 7(-1) + C \
   5 = 1 - 7 + C \
   5 = -6 + C \
   C = 11
   $$
   
3. **Write the particular solution:**
   $$
   y = x^2 + 7x + 11
   $$

4. **Find $$y$$ when $$x = 2$$:**
   $$
   y = (2)^2 + 7(2) + 11 = 4 + 14 + 11 = 29
   $$

**Answer:**
$$
y = 29 \quad \text{when} \quad x = 2
$$

---

### **Problem 7**

**Given:**
- $$\frac{dv}{dt} = 3 - 2t$$
- $$v = 2$$ when $$t = 1$$

**Find:**
- The value of $$v$$ when $$t = 2$$.

**Solution:**

1. **Integrate $$\frac{dv}{dt}$$ to find $$v$$:**
   $$
   \frac{dv}{dt} = 3 - 2t \implies v = \int (3 - 2t) \, dt = 3t - t^2 + C
   $$
   where $$C$$ is the constant of integration.

2. **Apply the initial condition $$v = 2$$ when $$t = 1$$ to find $$C$$:**
   $$
   2 = 3(1) - (1)^2 + C \
   2 = 3 - 1 + C \
   2 = 2 + C \
   C = 0
   $$

3. **Write the particular solution:**
   $$
   v = 3t - t^2
   $$

4. **Find $$v$$ when $$t = 2$$:**
   $$
   v = 3(2) - (2)^2 = 6 - 4 = 2
   $$

**Answer:**
$$
v = 2 \quad \text{when} \quad t = 2
$$

---

Both problems involve solving a first-order differential equation with an initial condition to find the particular solution, and then evaluating that solution at a specific point.
**Problem 6:** - When $$x = 2$$, $$y = 29$$. **Problem 7:** - When $$t = 2$$, $$v = 2$$.

Given that and when , find the value of when

Given that and when , find the value of when

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