BCA Semester 4 – Numerical Methods – Unit 5: Solution of Ordinary Differential Equations (7Hrs.)

Use the classical RK method to estimate y(0.4) of the equation

$\quad \quad \frac{dy}{dx}=x^2+y^2$

with y(0)=0, assume $h=0.2$.

Prepare the multi-step methods available for solving ordinary differential equations. Evaluate the value of y at $x = 0.1$ and $0.2$ to 4 decimal places given

$\frac{dy}{dx}=x2y−1,\quad y(0)=1,$

using Taylor series method.

Define ordinary differential equation. Use the fourth order Runge-Kutta method to estimate $y(0.4)$ of the equation:

$\quad \quad \frac{dy}{dx}=x2+y2$

with $y(0) = 0$ assuming that $h = 0.2$

a) Write and implement an algorithm to solve the system of linear equations using Gauss-Seidel method with suitable example
b) Write and implement an algorithm to solve the ODE using Heun's method.

Using Runge-Kutta method of 4th order solve the following equation taking each step $h = 0.1$

$\quad \quad \quad \frac{dy}{dx}=4xy - xy$

given $y(0) = 3.3$ calculate y at $x = 0.1\space and\space 0.2.$

Define initial value problems and final value problems. Using heun’s method, find value of y when $x=0.3$ given that

$\quad \quad \frac{dy}{dx} = x + y \space and \space y=1$

when $x=0$.