← 2013 Paper 2

UPSC 2013 Maths Optional Paper 2 Q7b — Step-by-Step Solution

15 marks · Section B

Euler's method (and modified Euler) · Numerical Analysis · asked 2× in 14 yrs · Read the full method →

Question

Use Euler’s method with step size $h=0.15$ to compute the approximate value of $y(0.6)$, correct up to five decimal places from the initial value problem

Technique

Standard forward Euler — explicit, first-order accurate.

Solution

Setup. $f(x,y)=xy+x^{2}-2$. Step size $h=0.15$. Number of steps: $(0.6-0)/0.15=4$.

Euler’s recurrence: $y_{n+1}=y_n+h\,f(x_n,y_n)$.

Iterations

Sample computations

Step 1 → Step 2 ($x_1=0.15$, $y_1=1.70000$):

• $f(0.15,1.70)=0.15(1.70+0.15)-2=0.15(1.85)-2=0.2775-2=-1.72250$.
• $y_2=1.70000+0.15(-1.72250)=1.70000-0.25838=1.44163$ (rounded to 5 dp).

Step 2 → Step 3 ($x_2=0.30$, $y_2=1.44163$, using full precision $1.441625$):

• $f(0.30,1.441625)=0.30(1.741625)-2=0.5224875-2=-1.4775125$.
• $y_3=1.441625-0.221627=1.219998$ (5 dp).

Step 3 → Step 4 ($x_3=0.45$, $y_3=1.219998$):

• $f(0.45,1.219998)=0.45(1.669998)-2=0.751499-2=-1.248501$.
• $y_4=1.219998-0.187275=1.032723$.

Final answer

Answer