Question

Let \( y(x) \) be the solution of the initial value problem \[ \frac{dy}{dx} = \sin(\pi(x + y)), \quad y(0) = 0. \] Using Euler's method, with the step-size \( h = 0.5 \), the approximate value of \( y(1.5) + 2y(1) \) is equal to (in integer):

Hint:

Use Euler's method with a fixed step-size to approximate the solution of an initial value problem. At each step, compute the function's value and update the solution iteratively.

Answer 1

Step 1: Applying Euler's Method
Euler's method is given by the iterative formula:

\[ y_{n+1} = y_n + h f(x_n, y_n) \]
where \( h \) is the step-size and \( f(x, y) = \sin(\pi(x + y)) \).

Starting with \( y(0) = 0 \), we compute the values of \( y \) at each step.

Step 2: Compute \( y(0.5) \)
At \( x = 0 \), \( y_0 = 0 \). Using the formula:
\[ y_1 = y_0 + h \sin(\pi(0 + y_0)) = 0 + 0.5 \sin(0) = 0 \]
So, \( y(0.5) = 0 \).

Step 3: Compute \( y(1) \)
At \( x = 0.5 \), \( y_1 = 0 \). Using the formula:
\[ y_2 = y_1 + h \sin(\pi(0.5 + y_1)) = 0 + 0.5 \sin(\pi \times 0.5) = 0 + 0.5 \times 1 = 0.5 \]
So, \( y(1) = 0.5 \).

Step 4: Compute \( y(1.5) \)
At \( x = 1 \), \( y_2 = 0.5 \). Using the formula:
\[ y_3 = y_2 + h \sin(\pi(1 + y_2)) = 0.5 + 0.5 \sin(\pi \times 1.5) = 0.5 + 0.5 \times (-1) = 0 \]
So, \( y(1.5) = 0 \).

Step 5: Calculate the Final Result
We need to compute \( y(1.5) + 2y(1) \):
\[ y(1.5) + 2y(1) = 0 + 2(0.5) = 1 \]
Thus, the approximate value is \( \boxed{1} \).

\[ \boxed{1} \]