numerical methods

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MA1251 – NUMERICAL METHODS
UNIT – IV : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL
EQUATIONS
PART – A
1. By Taylor series, find y(1.1) given y = x + y, y(1) = 0.
2. Find the Taylor series upto x3 term satisfying 2y y x 1, y(0) 1.
3. Using Taylor series method find y at x = 0.1 if 1, (0) 1 2 x y y
dx
dy
.
4. State Adams – Bashforth predictor and corrector formula.
5. What is the condition to apply Adams – Bashforth method ?
6. Using modified Euler’s method, find y(0.1) if , (0) 1 2 2 y x y
dx
dy
.
7. Write down the formula to solve 2nd order differential equation using Runge-
Kutta method of 4th order.
8. In the derivation of fourth order Runge-Kutta formula, why is it called fourth
order.
9. Compare R.K. method and Predictor methods for the solution of Initial value
problems.
10. Using Euler’s method find the solution of the IVP log(x y), y(0) 2
dx
dy
at x 0.2 taking h 0.2.
PART-B
11. The differential equation
2 y x
dx
dy
is satisfied
by y(0) 1, y(0.2) 1.12186, y(0.4) 1.46820, y(0.6) 1.7379.Compute the value
of y(0.8) by Milne’s predictor - corrector formula.
12. By means of Taylor’s series expension, find y at x = 0.1,and x = 0.2 correct to
three decimals places, given x y e
dx
dy
2 3 , y(0) = 0.
13. Given y xy y 0, y(0) 1, y (0) 0, find the value of y(0.1) by using
R.K.method of fourth order.
14. Using Taylor;s series method find y at x = 0.1, if 1 2 x y
dx
dy
, y(0)=1.
15. Given (1 ) 2 x y
dx
dy
, y(1) = 1, y(1.1) = 1.233, y(1.2) = 1.548, y(1.3)=1.979,
evaluate y(1.4) by Adam’s- Bashforth method.
16. Using Runge-Kutta method of 4th order, solve
2 2
2 2
y x
y x
dx
dy
with y(0)=1 at
x=0.2.
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17. Using Milne’s method to find y(1.4) given that 5 2 0 2 xy y given that
y(4) 1, y(4.1) 1.0049, y(4.2) 1.0097, y(4.3) 1.0143.
18. Given
, (0) 2, (0.2) 2.443214, (0.4) 2.990578, (0.6) 3.823516 3 x y y y y y
dx
dy
find y(0.8) by Milne’s predictor-corrector method taking h = 0.2.
19. Using R.K.Method of order 4, find y for x = 0.1, 0.2, 0.3 given that
, (0) 1 2 xy y y
dx
dy
also find the solution at x = 0.4 using Milne’s method.
20. Solve 2 y x
dx
dy
, y(0) = 1.
Find y(0.1) and y(0.2) by R.K.Method of order 4.
Find y(0.3) by Euler’s method.
Find y(0.4) by Milne’s predictor-corrector method.
21. Solve 0.1(1 ) 0 2 y y y y subject to y(0) 0, y (0) 1 using fourth order
Runge-Kutta Method.
Find y(0.2) and y (0.2) . Using step size x 0.2.
22. Using 4th order RK Method compute y for x = 0.1 given
2 1 x
xy
y given y(0) =
1 taking h=0.1.
23. Determine the value of y(0.4) using Milne’s method given , (0) 1 2 xy y y
dx
dy
,
use Taylors series to get the value of y at x = 0.1, Euler’s method for y at x = 0.2
and RK 4th order method for y at x=0.3.
24. Consider the IVP 1, (0) 0.5 2 y x y
dx
dy
(i) Using the modified Euler method, find y(0.2).
(ii) Using R.K.Method of order 4, find y(0.4) and y(0.6).
(iii) Using Adam- Bashforth predictor corrector method, find y(0.8).
25. Consider the second order IVP 2 2 int, 2 y y y e S t with y(0) = -0.4 and
y’(0)=-0.6.
(i) Using Taylor series approximation, find y(0.1).
(ii) Using R.K.Method of order 4, find y(0.2).