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

End

-Copyrights Reserved-

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

End

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