← 2019 Paper 2

UPSC 2019 Maths Optional Paper 2 Q5d — Step-by-Step Solution

10 marks · Section B

Runge-Kutta methods (RK2/RK4) · Numerical Analysis · asked 4× in 13 yrs · Read the full method →

Question

Using Runge-Kutta method of fourth order, solve $\dfrac{dy}{dx}=\dfrac{y^2-x^2}{y^2+x^2}$ with $y(0)=1$ at $x=0.2$. Use four decimal places for calculation and step length $0.2$.

Technique

Single fourth-order Runge–Kutta step.

Solution

Setup. $f(x,y)=\dfrac{y^2-x^2}{y^2+x^2}$, $x_0=0$, $y_0=1$, $h=0.2$. One RK4 step reaches $x=0.2$.

RK4:

Step 1 — $k_1$

$f(0,1)=\dfrac{1-0}{1+0}=1$.

Step 2 — $k_2$ at $(0.1,\,1+0.1=1.1)$

$f(0.1,1.1)=\dfrac{1.21-0.01}{1.21+0.01}=\dfrac{1.20}{1.22}=0.983607$.

Step 3 — $k_3$ at $(0.1,\,1+0.098361=1.098361)$

$f(0.1,1.098361)=\dfrac{1.206397-0.01}{1.206397+0.01}=\dfrac{1.196397}{1.216397}=0.983559$.

Step 4 — $k_4$ at $(0.2,\,1+0.196712=1.196712)$

$f(0.2,1.196712)=\dfrac{1.432119-0.04}{1.432119+0.04}=\dfrac{1.392119}{1.472119}=0.945656$.

Step 5 — Combine

Answer