North Maharashtra University 2008 B.Sc Mathematics S.YBSc - MTH – 222 (B) (Numerical Analysis) - Question Paper

2) define the Euler’s method of finding the solution of differential
formula y' = f(x,y) with y(x0) = y0 .
3) define the Euler’s replaced method of finding the solution of
differential formula y' = f(x,y) with y(x0) = y0 .
23
4) Why Runge-Kutta methods are more effective in finding the solution of
differential formula y' = f(x,y) with y(x0) = y0 . discuss Runge-Kutta
second order formulae.
5) State the Runge-Kutta method 4th order formulae for finding the
solution of differential formula y' = f(x,y) with y(x0) = y0 . obtain y(0.1) by
Runge-Kutta 2nd order formulae where y' = y – x with y(0) = two and h
= 0.1.
6) Using the Taylor’s series for y(x), obtain y(0.1) accurate to 4 decimal
places if y(x) satisfies y' = x + (–y2) with y(0) = 1.
7) Solve the differential formula y' = x + y with y(0) = 1, x ? [0 , 1] by
Taylor’s series expansion to find y for x = 0.1.
8) Using Taylor’s series expansion, obtain the solution of the differential
formula y' = (0.1) (x3 + y2) with y(0) = one accurate to four decimal places.
9) Using Taylor’s method, find y(1.3) if the differential formula is y' =
x2 + y2 with y(1) = 0.
10) Using Taylor’s method, find y(0.1) provided that y' = xy + one with y(0) = 1.
11) Using Taylor’s method, find y(4.1) and y(4.2), provided that y' =
x y
1
2 +
with y(4) = 4.
12) find the Taylor’s series for the differential formula y' = ysinx + cosx
with y(0) = 0.
13) obtain y(1.2) by Taylor’s series for y(x) provided that y' = x + y with y(1) = 0.
14) Using Euler’s method, solve the differential formula y' =
y x
y x
+
- , y(0) =
1, obtain y(0.1) in four steps.
15) Using Euler’s method, obtain y(0.5), provided that y' = y2 – x2 with y(0) = 1
and h = 0.1.
16) Using Euler’s method, obtain y(1.5), provided that y' = xy with y(1) = five in the
interval [1 , 1.5] and h = 0.1.
24
17) Using Euler’s method, obtain y(0.2), y(0.4) provided that y' =
y x
y x
+
- with y(0)
= one and h = 0.1.
18) Use Euler’s method for every to calculate
i) y(0.1) and y(0.2), provided that y' + 2y = 0 with y(0) = one and h = 0.1.
ii) y(0.1) and y(0.2), provided that y' = one + y2 with y(0) = one and h = 0.1.
iii) y(0.02) and y(0.03), provided that y' = –y with y(0) = one and h = 0.01.
iv) y(0.4) and y(0.6), provided that y' = x + y with y(0) = 0 and h = 0.2.
v) y(0.4) provided that y' = xy with y(0) = one and h = 0.1.
vi) y(2), provided that y' = xy + two with y(1) = one and h = 0.1.
vii) y(0.5), provided that y' = x2 + y2, with y(0) = 0 and h = 0.1.
19) Use Euler’s method for every to calculate
i) y(0.2) and y(0.4), provided that y' = x + xy with y(0) = one and h =
0.2.
ii) y(0.5) and y(0.1), provided that y' = x + y with y(0) = one and h = 0.05.
iii) y(0.2), provided that y' = log10(x + y) with y(0) = one and h = 0.2.
iv) y(0.1), provided that y' = x2 + y with y(0) = one and h = 0.05.
v) y(0.02) and y(0.04), provided that y' = x2 + y with y(0) = one and h =
0.01.
20) Using Runge-Kutta 2nd order formulae calculate y(0.1) and y(0.2)
accurate to 4 decimal places, provided that y' = y – x with y(0) = 2
and h = 0.1.
21) Using Runge-Kutta 4th order formulae calculate
i) y(0.1), provided that y' = 3x +
2
y with y(0) = 1at x = 0.1 and h =
0.1.
ii) y(0.2), provided that y' = –xy with y(0) = one and h = 0.2.
iii) y(0.2), with y' = x + y with y(0) = one and h = 0.1.
iv) y(1.1), provided that y' = xy1/3 with y(1) = one and h = 0.1.
25
v) y(0.4), provided that y' = –2xy2 with y(0) = one and h = 0.2.
vi) y(0.1), provided that y' = y – x with y(0) = two and h = 0.1.
vii) y(0.2) and y(0.4), provided that y' = one + y2 with y(0) = 0 and h =
0.2.
viii) y(1), provided that y' =
y x
y x
+
- with y(0) = one and h = 0.05.
ix) y(1.4), provided that y' = xy with y(1) = two and h = 0.2.

Earning:   Approval pending.