COMSATS UNIVERSITY ISLAMABAD
Department of Electrical Engineering (EE-A)
Ordinary Differential Equations MATH-241
Max Marks 50
Time Allowed: 03 Hour
INSTRUCTIONS
Attempt all questions with complete detail; marks of each questions are duly
mentioned.
Write your name and registration ID on all answer sheets.
Write page number on all sheets and Image must be readable
Make single Pdf file of answer sheets and rename the file with your name & registration
ID.
Upload single Pdf file on MS-TEAM within the given time. Late submission will
reduce the marks 20%.
Please be analytical and specific in answering questions.
The students should retain the answer sheet until the announcement of results.
Q 1: The differential equation (𝑒 𝑥 𝑠𝑒𝑐𝑦 − 𝑡𝑎𝑛𝑦) + 𝑦′ = 0, has an integrating factor of
the form 𝑒 −𝛽𝑥 𝑐𝑜𝑠𝑦 for some constant 𝛽. Find 𝛽 and then solve the differential
equation. [10]
Q 2: Solve the following differential equation with the help of Laplace Transformation
𝑦 ′′′ − 6𝑦 ′′ + 11𝑦 ′ − 6𝑦 = 𝑒 4𝑡 ; 𝑦(0) = 𝑦 ′ (0) = 𝑦 ′′ (0) = 0. [10]
Q 3: Solve the given initial-value problem. Does the solution of the differential equation
is linearly independent? Also verify the result.
𝑦 ′′′ + 12𝑦 ′′ + 36𝑦 ′ = 0; 𝑦(0) = 0, 𝑦 ′ (0) = 1, 𝑦 ′′ (0) = −7. [10]
Q 4: Solve the following differential equation using variation of parameters.
𝑦 ′′ + 2𝑦′ + 𝑦 = 𝑒 −𝑥 𝑙𝑛𝑥. [10]
Q 5: Find the ‘’recurrence relation’’ of the given differential equation. Also find two
linearly independent solutions of the differential equation
1 1
𝑥 2 𝑦 ′′ + 𝑥 ( + 2𝑥) 𝑦 ′ + (𝑥 − ) 𝑦 = 0. [10]
2 2
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