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Tutorial 3+4

This document summarizes topics covered in a tutorial on solving second order ordinary differential equations, including: 1. Using the Wronskian to determine linear dependence or independence of solution sets. 2. Verifying solutions and checking their linear independence using the Wronskian. 3. Solving homogeneous second order ODEs. 4. Solving second order ODEs with initial-value conditions. 5. Obtaining general solutions to nonhomogeneous ODEs using the method of undetermined coefficients. 6. Obtaining general solutions to other nonhomogeneous ODEs using undetermined coefficients. 7. Solving nonhomogeneous ODEs with initial values using undetermined coefficients. 8

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42 views2 pages

Tutorial 3+4

This document summarizes topics covered in a tutorial on solving second order ordinary differential equations, including: 1. Using the Wronskian to determine linear dependence or independence of solution sets. 2. Verifying solutions and checking their linear independence using the Wronskian. 3. Solving homogeneous second order ODEs. 4. Solving second order ODEs with initial-value conditions. 5. Obtaining general solutions to nonhomogeneous ODEs using the method of undetermined coefficients. 6. Obtaining general solutions to other nonhomogeneous ODEs using undetermined coefficients. 7. Solving nonhomogeneous ODEs with initial values using undetermined coefficients. 8

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Omar A-g
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KIX 1002: ENGINEERING MATHEMATICS 2

Tutorial 3 – 4: 2nd Order Ordinary Differential Equations


1. Use the Wronskian to show whether the give set of functions is linearly dependent or
linearly independent.
(a) 𝑦1 = 𝑥 𝑦2 = 𝑥 + 1
𝛼𝑥
(b) 𝑦1 = 𝑒 sin 𝛽𝑥 𝑦2 = 𝑒 𝛼𝑥 cos 𝛽𝑥

2. Verify that each of the given functions is a solution of the differential equation, and use
their Wronskian to show that these solutions are linearly independent.
Verify the linear combination of the solutions is also a solution.
(a) 𝑦1 = 𝑒 𝑥 𝑦2 = 𝑒 −2𝑥 𝑦" + 𝑦′ − 2𝑦 = 0
2 −1
(b) 𝑦1 = 𝑥 𝑦2 = 𝑥 𝑥 2 𝑦" − 2𝑦 = 0
(c) 𝑦1 = 𝑒 −𝑥 𝑦2 = 𝑥𝑒 −𝑥 𝑦" + 2𝑦′ + 𝑦 = 0

3. Solve the following second order ODEs


(a) 𝑦" − 4𝑦 = 0
(b) 𝑦" − 𝑦′ − 6𝑦 = 0
(c) 6𝑦" + 𝑦′ − 𝑦 = 0
(d) 𝑦" + 𝑦 = 0
(e) 𝑦" + 4𝑦′ + 8𝑦 = 0

4. Solve the following second order ODEs with initial-value conditions


(a) 𝑦" − 4𝑦 = 0 𝑦(0) = 4 𝑦′(0) = 12
(b) 𝑦" − 6𝑦 + 9𝑦 = 0 𝑦(0) = −1.4 𝑦′(0) = 4.6
1 1
4 2
(c) 4𝑦" − 8𝑦 + 3𝑦 = 0 𝑦(1) = 5 𝑒 2 𝑦′(1) = 5 𝑒 2
(d) 𝑦" + 𝑦 = 0 𝑦(0) = 3 𝑦′(0) = −0.5
(e) 𝑦" + 2𝑦′ + 2𝑦 = 0 𝑦(0) = 1 𝑦′(0) = −1

5. Obtain a general solution for nonhomogeneous problems using the method of


undetermined coefficients.
(a) 𝑦" + 3𝑦′ + 2𝑦 = 2𝑥 2
(b) 𝑦" + 4𝑦 = 4𝑥 2 + 6
(c) 𝑦" − 𝑦 = 𝑒 𝑥
(d) 𝑦" + 3𝑦′ − 4𝑦 = 10𝑒 𝑥
(e) 𝑦" − 4𝑦′ + 3𝑦 = 2𝑒 3𝑥
(f) 𝑦" + 4𝑦′ − 2𝑦 = 2𝑥 2 − 3𝑥 + 6
(g) 𝑦" + 3𝑦′ + 2𝑦 = 6
6. Obtain a general solution for nonhomogeneous problems using the method of
undetermined coefficients.
(a) 𝑦" − 2𝑦′ + 5𝑦 = 𝑒 𝑥 cos 2𝑥
(b) 𝑦" + 𝑦 = 2𝑥 sin 𝑥
(c) 𝑦" + 2𝑦′ + 2𝑦 = 𝑒 𝑥 sin 𝑥
(d) 𝑦" − 3𝑦′ + 2𝑦 = 𝑥 2 𝑒 3𝑥

7. Solve the following nonhomogeneous with initial value using the method of
undetermined coefficients.
(a) 5𝑦" + 𝑦′ = −6𝑥 𝑦(0) = 0 𝑦 ′ (0) = 0
(b) 𝑦" + 4𝑦′ + 5𝑦 = 35𝑒 −4𝑥 𝑦(0) = −3 𝑦 ′ (0) = 1
(c) 𝑦" + 25𝑦 = 5𝑥 𝑦(0) = 5 𝑦 ′ (0) = −4.8
(d) 𝑦" − 2𝑦′ + 𝑦 = 2𝑥 2 − 8𝑥 + 4 𝑦(0) = 0.3 𝑦′(0) = 0.3
2𝑥
(e) 𝑦" − 𝑦′ − 2𝑦 = 3𝑒 𝑦(1) = 𝑒 −1 𝑦 ′ (1) = −𝑒 −1 + 𝑒 2

8. Obtain a general solution using the method of variation of parameter.


(a) 𝑦" − 4𝑦′ + 4𝑦 = 𝑒 2𝑥 /𝑥
(b) 𝑦" + 2𝑦′ + 𝑦 = 𝑥 3/2 /𝑒 𝑥
(c) 𝑦" − 2𝑦′ + 𝑦 = 𝑒 𝑥 sin 𝑥
(d) 𝑦" + 2𝑦′ + 2𝑦 = 𝑒 −𝑥 sec 𝑥
(e) 𝑦" − 4𝑦′ + 4𝑦 = (𝑥 + 1)𝑒 2𝑥
(f) 𝑦" − 5𝑦′ + 6𝑦 = 2𝑒 𝑥
(g) 𝑦" + 2𝑦′ + 𝑦 = 4𝑒 −𝑥

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