7 Given that $$ y(1)=-3 $$ and $$ \frac{d y}{d x}=2 x+y $$, what is the approximation for $$ y(2) $$ if Euler's method is used with a step size of 0.5 , starting at $$ x=1 $$ ?

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To approximate $$ y(2) $$ using Euler's method with a step size of $$ h = 0.5 $$, starting from $$ x = 1 $$ where $$ y(1) = -3 $$, we proceed as follows:

Euler's method updates the solution using the formula:
$$
y_{n+1} = y_n + h \cdot f(x_n, y_n)
$$
where $$ f(x, y) = \frac{dy}{dx} $$.

Given:
$$
f(x, y) = 2x + y
$$

**Step 1: From $$ x = 1 $$ to $$ x = 1.5 $$**

- **Initial point:** $$ (x_0, y_0) = (1, -3) $$
- **Compute $$ f(x_0, y0) $$:**
  $$
  f(1, -3) = 2(1) + (-3) = 2 - 3 = -1
  $$
- **Update $$ y $$:**
  $$
  y_1 = y_0 + h \cdot f(x_0, y_0) = -3 + 0.5 \cdot (-1) = -3.5
  $$
- **New point:** $$ (x_1, y_1) = (1.5, -3.5) $$

**Step 2: From $$ x = 1.5 $$ to $$ x = 2 $$**

- **Current point:** $$ (x_1, y_1) = (1.5, -3.5) $$
- **Compute $$ f(x_1, y1) $$:**
  $$
  f(1.5, -3.5) = 2(1.5) + (-3.5) = 3 - 3.5 = -0.5
  $$
- **Update $$ y $$:**
  $$
  y_2 = y_1 + h \cdot f(x_1, y_1) = -3.5 + 0.5 \cdot (-0.5) = -3.75
  $$
- **Final approximation:** $$ y(2) \approx -3.75 $$

**Conclusion:**
Using Euler's method with a step size of 0.5, the approximation for $$ y(2) $$ is **$$-3.75$$**.

**Answer:**  
After two Euler steps, an approximation for y ( 2 ) is –3.75
After applying Euler's method with a step size of 0.5, the approximation for $$ y(2) $$ is **–3.75**.

7 Given that \( y(1)=-3 \) and \( \frac{d y}{d x}=2 x+y \), what is the approximation for \( y(2) \) if Euler's method is used with a step size of 0.5 , starting at \( x=1 \) ?

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