Department of Mathematics and Philosophy of Engineering
MHZ3531 Engineering Mathematics IA
Assignment No: 2
Academic Year: 2020/2021 Due Date: Will be notify later
Instructions
• Answer all the questions.
• Attach the cover page with your answer scripts.
• Use both sides of papers when you are answering the assignment.
• Please send the answer scripts of your assignment on or before the due date to the following address.
Course Coordinator – MHZ3531,
Dept. of Mathematics & Philosophy of Engineering,
Faculty of Engineering Technology,
The Open University of Sri Lanka,
P.O. Box 21,
Nawala,
Nugegoda.
MHZ3531-Engineering Mathematics IA
Assignment 2
Q1 (a) Write down the order and the degree of the following differential equations.
i) ???? ???? + (???? + sin ??) = 0 ii) ??2?? 3 ???? 4 3
iii) ??3?? 5 ??2?? iv) ??2?? 3
v)
(b) Using the method of variable separation, solve the following differential equations
i) (???? + ??)???? = (??2??2 + ??2+??2 + 1)????
ii)
iii)
iv) ??????2?? ???? = ??????2?? ????
v)
Q2 (a) Solve the following homogeneous differential equations.
i) ???? ???? ii)
iii) ???? iv)
b) Show that the following differential equations are exact and solve them.
i)
ii) (?? + sin ??)???? + (?? cos ?? – 2??)???? = 0 iii) 2?? + ?? cos ????)???? + ?? cos ???? ???? = 0
(c) Using an integrating factor, solve the following differential equations.
i) (??2 + ??2 + ??)???? + ???? ???? = 0
ii)
.
Q3
?? = ??????, then show that ?? has two distinct real values. Further, if the values of ?? are ??1 and ??2, then show that ?? = ??1????1?? + ??2????2?? is the complete primitive of the given differential equation, where ??1 and ??2 are arbitrary constants.
?? = ??????, then show that ?? has two distinct imaginary values. Further, show that the complete primitive of the above differential can be expressed of the form
?? = ??????(??1 cos ???? + ??2 sin ????), where ??1 and ??2 are arbitrary constants.
Q4 (a) Let ??(??) = ??2 – 3. Show that the equation ??(??) = 0 has a root between 1 and 2.
i) By using the bisection method, find a solution for the above equation correct to nine decimal places.
ii) By applying Newton Raphson’s method, find a solution for above equation correct to nine decimal places taking ??0 = 1.5.
(b) Using Newton’s interpolation divided difference formula and the following table calculate an approximation value for ??(1.5).
???? -2 0 1 4 5
??(????) -8 0 1 64 125
(c) Using Lagrange’s interpolation formula and the following table, calculate an approximation value for ??(10).
???? 5 6 9 11
??(????) 12 13 14 16
where ?? interpolates ?? at these points.
Q5 (a) Write down Trapezoidal Rule and Simpson’s rule to approximate the finite
.
i) Using the Trapezoidal Rule with ?? = 6 subintervals, approximate the integral
approximation.
ii) Using Simpson’s Rule with ?? = 4 subintervals, approximate the
integral ? ?? ?? ???? to 3 decimal places. Estimate the relative percent
-2
error of the approximation.
(b) Using the Jacobi’s iteration method, find the sixth iteration of the solution of the following system of equations.
3?? + 10?? – ?? = -8
2?? – 3?? + 10?? = 15
10?? + ?? – 2?? = 7
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