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

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To find the value of $$ y $$ when $$ x = 2 $$, follow these steps:

1. **Given Differential Equation:**
   $$
   \frac{dy}{dx} = 2x + 7
   $$

2. **Integrate to Find $$ y $$:**
   $$
   y = \int (2x + 7) \, dx = x^2 + 7x + C
   $$
   where $$ C $$ is the constant of integration.

3. **Apply the Initial Condition $$ y = 5 $$ when $$ x = -1 $$:**
   $$
   5 = (-1)^2 + 7(-1) + C \
   5 = 1 - 7 + C \
   5 = -6 + C \
   C = 11
   $$
   
4. **Write the Particular Solution:**
   $$
   y = x^2 + 7x + 11
   $$

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

**Answer:** $$ y = 29 $$ when $$ x = 2 $$.
When $$ x = 2 $$, $$ y = 29 $$.

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

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