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: Set up the Euler table directly
Instead of computing point-by-point conceptually, we tabulate Euler’s method. Given:

 

• Differential equation: \( y' = \sin(\pi(x+y)) \)
• Initial condition: \( y(0)=0 \)
• Step size: \( h = 0.5 \)

Euler’s update rule is: 

\[ y_{n+1} = y_n + 0.5\,\sin\!\big(\pi(x_n+y_n)\big) \]

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Step 2: Build values iteratively in tabular form

\(n\)	\(x_n\)	\(y_n\)	\(\sin(\pi(x_n+y_n))\)	\(y_{n+1}\)
0	0	0	\(\sin(0)=0\)	\(0\)
1	0.5	0	\(\sin(\pi/2)=1\)	\(0.5\)
2	1	0.5	\(\sin(3\pi/2)=-1\)	\(0\)

Thus:

• \(y(0.5)=0\)
• \(y(1)=0.5\)
• \(y(1.5)=0\)

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Step 3: Compute the required expression

\[ y(1.5)+2y(1)=0+2(0.5)=1 \]

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Final Answer

\[ \boxed{1} \]