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NMCP Unit 6

This document contains 25 questions related to numerical methods for solving ordinary differential equations. The questions cover topics like Taylor series methods, Euler's method, Runge-Kutta methods, and numerical integration techniques including the trapezoidal rule, Simpson's rule, and Newton-Cotes formulas. The questions ask students to use these numerical methods to solve sample differential equations, estimate integrals, and evaluate errors.

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Shashikant Prasad
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209 views3 pages

NMCP Unit 6

This document contains 25 questions related to numerical methods for solving ordinary differential equations. The questions cover topics like Taylor series methods, Euler's method, Runge-Kutta methods, and numerical integration techniques including the trapezoidal rule, Simpson's rule, and Newton-Cotes formulas. The questions ask students to use these numerical methods to solve sample differential equations, estimate integrals, and evaluate errors.

Uploaded by

Shashikant Prasad
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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Question Bank

Unit-6

Q1. Use Taylor method recursively to solve the equation


y ' =x 2 + y 2 with y ( 0 )=0 for the interval (0.04) using two
subintervals of size 0.2.
Q2. Find out the solution of y ' =2 y +3 e x using Taylor’s series.
Initial values are given as x 0=0∧ y 0=1.Find out value of y for
0(0.1) 0.3.
Q3. Solve the following equation by Taylor series method :
x y =x − y Given y (2)=2 Find at y at x=2.1
'

dy 2
Q4. Solve dx
=x− y by Taylor’s series method to calculate y at
x=0.4 in two steps. Initial values are x=0,y=1.
Q5. Find 1(0.1) 1.2 [Intial value=1,h=0.1 Final value =1.2],the
dy
solution of dx
=x+ y .

Q6. Explain Taylor’s series method for solution of ordinary


differential equations.
Q7. Using Euler’s method , obtain the solution of y ' =x− y ,given
x 0=0 , y 0=1 at x =0.6 taking h=0.2.
Q8. Using Euler’s method ,find an approximate value of y
corresponding to x=1, y ' =x + y∧ y =1 when x=0.
Q9. Use Euler’s method to numerically integrate,
f ( x , y )=−2 x3 +12 x 2−20 x+ 8.5 , y ( 0 )=1¿ x=0 ¿ x=0.5 Giving proper
example illustrate the effect of stop size on stability of Euler’s
method.
Q10. Using Euler’s method solve the following equation
dy
=x+ y , y ( 0 )=0 choose h=0.2 and compute y(0.4) and y(0.6).
dx
Q11. Use the predictor –corrector formula for tabulating the
dy 2 2
solutions of 10 dx = x + y , y ( 0 )=1 for the range 0.5 ≤ x ≤ 1.0.

Q12. Explain modified Euler’s method for solution of ordinary


differential equations.
Q13. Use 4th order Rk method to estimate y(0.4) when y ' =x 2 + y 2
with y(0)=0 Take h=0.2.
dy
Q14. Solve dx
=x+ y when y=1 at x =0. Find solution for x=0.1,0.2 by
Runge-kutta method.
Q15. Solve the following equation by Runge-kutta method at
dy
x=0.8: dx
= y−x Take x 0=0 , y ( 0 )=2, h=0.2.
dy
Q16. Solve the equation dx =√ x+ y , subject to x=0,y=1 to find y
at x=0.2 taking h=0.1.
π

Q17. Evaluate the interal ∫ (4 +2 sin x ) dx using Simpson,s


0
3th
rule wheren=5. Compute percent relative error.
8
Q18. Use trapezoidal rule with four steps to estimate the rule of
2
x
integral. ∫ dx.
0 √ 2+ x2
5.2

Q19. Evaluate ∫ ¿ x dx using trapezoidal rule.Take h=0.2.


4
1
sinx
Q20. Sove ∫ x
dx using trapezoidal rule and write a program
0

for this same.


Q21. Derive the formula of traperzoidal rule as special case of
Newton cotes’s formulae for numerical integration.
1rd
Q22. Derive formula of Simpson’s 3 rule using Newton cote’s
formula for numerical integration.
Q23. A curve is drawn to pass through the points given by the
following table.
x 1 1.5 2 2.5 3 3.5 4
y 2 2.4 2.7 2.8 3 2.6 2.1
Estimate the area bouned by the curve,the x-axis and ordinates
x=1,x=4.
π
2 3th
Q24. Evaluate ∫ e sinx dx , using simpson’s 8 rule with 6 sub
0

intervals.
π

Q 25.Evaluate the interal ∫ (4 +2 sin x ) dx using Simpson,s


0
3th
rule wheren=5. Compute percent relative error.
8

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