School of Mathematics, Thapar University, Patiala

UMA032 : Numerical and Statistical Methods

Assignment 7

Numerical Differential Equations

1. Consider the IVPy′ = x(y + x)− 2, y(0) = 2.

Use the Euler method with stepsize h = 0.2 to compute y(0.6) with four decimals.

2. Use modified Euler’s method to find y(0.2) and y(0.4) with h = 0.2 for IVP

y′ = y + ex, y(0) = 0.

3. Solve the initial value problemdy

dx=

y + x2 − 2

x + 1, y(0) = 2

by explicit Euler method with step size h = 0.2 for interval [0, 1].

4. Solve the following initial value problem with step size h = 0.1 and 0.2.

y′ = te−t − y, y(0) = 1

by explicit Euler method in the interval [0, 1].

5. Solve the following differential equation by second-order Runge-Kutta method

y′ = −y + 2 cos t, y(0) = 1.

Compute y(0.2), y(0.4), and y(0.6).

6. Compute solutions to the following problems with a second-order Taylor method. Use step size h = 0.2.

(A)

y′ = (cos y)2, 0 ≤ x ≤ 1, y(0) = 0.

(B)

y′ =20

1 + 19e−x/4, 0 ≤ x ≤ 1, y(0) = 1.

7. Using Runge-Kutta fourth-order method to solve the IVP at x = 0.8 for

dy

dx=√x + y, y(0.4) = 0.41

with step length h = 0.2.

8. Use the Runge-Kutta fourth-order method to solve the following IVP

y′ = xz + 1, y(0) = 0,

z′ = −xy, z(0) = 1

with h = 0.1 and 0 ≤ x ≤ 0.2.

9. Apply the Taylor’s method of order three to obtain approximate value of y at x = 0.2 for the differentialequation

y′ = 2y + 3ex, y0 = 0.

Compare the numerical solution with the exact solution.

CONTINUED

– 2 –

10. Use Runge-Kutta method of order four to solve

y′′ = xy′2 − y2, y(0) = 1, y′(0) = 0

for x = 0.2 with stepsize 0.2.

11. Consider the Lotka-Volterra system

du

dt= 2u− uv, u(0) = 1.5

dv

dt= −9v + 3uv, v(0) = 1.5.

Use Euler’s method with step size 0.5 to approximate the solution at t = 2.

12. The following system represent a much simplified model of nerve cells

dx

dt= x + y − x3, x(0) = 0.5

dy

dt= −x

2, y(0) = 0.1

where x(t) represents voltage across the boundary of nerve cell and y(t) is the permeability of the cell wallat time t. Solve this system using Runge-Kutta fourth-order method to generate the profile up to t = 0.2with step size 0.1.