**ERG2011A Advanced Engineering Mathematics (Syllabus A),**

**Problem Set 2**

**Kreyszig 9th edition - 2.1 - 2.8, 2.10, Class Notes 2**

**1. (Linear independence)**

Which of the following sets of functions are linearly independent? Prove your assertion.

**2. (ADVANCED TOPIC QUESTION) (Reduction of order of ODEs)**

(a) Use the method of reduction of order discussed in class to reduce to first order and solve Kreyszig 9th edition, Problem Set 2.1, problems 20 and 22.

(b) (Second solution of Euler-Cauchy equation with repeated roots). Consider the ODE

As shown in class, is a solution to this ODE, for satisfying

If , then prove that there are two distinct values and that satisfy (1). On the other hand, if , use the method of reduction of order derived in class to prove that the two solutions equal and .

(c) Kreyszig 9th edition, Problem Set 2.1, problem 15, suggests using the substitution , and therefore , to reduce the order of ODEs in which functions of only , , and appear (but NOT of ). Use this observation to solve problems 17 and 18 from the same problem set.

(d) Kreyszig 9th edition, Problem Set 2.1, problem 16, also suggests using the substitution . However, in this case, it is then suggested that the chain rule be used to express as (prove this!). In this case, the problem suggests the reduction of the order of ODEs in which functions of only , , and appear (but NOT of ). Use this observation to solve problems 19 and 21 from the same problem set.

Note: The above two substitutions are important for general classes of 2nd order ODEs.

**3. (2nd order linear homogeneous ODEs)**

(a) Kreyszig 9th edition, Problem Set 2.2, Problems 21, 22, 23, 15, 20, 18. (Try them in that order)

Solve the initial value problem. Check that your answer satises the ODE as well as the initial conditions. (Show the details of your work.)

Find an ODE for the given basis.

(b) Consider an object undergoing damped simple harmonic motion so that its trajectory is of the form . Compute its velocity as a function of time. In terms of and , at what times do the local maxima of the object's velocity occur? Suppose you are told that for each , second, and further that the velocity at is half that at , what are the values of and ?

(c) Kreyszig 9th edition, Problem Set 2.5, Problems 2, 3, and 4.

Find a real general solution, showing the details of your work.

**4. (Linear independence: Wronskian)**

Kreyszig 9th Edition Problem set 2.6, problems 1, 2, 3, 12, 14.

Show linear independence by Wronskian.

**5. (Non-homogeneous ODEs: Method of undetermined coefficients)**

Kreyszig 9th Edition Problem set 2.7, problems 1, 2, 4, 5, 6, 12, 14.

Find a (real) general solution. (Show each step of your calculation.)

**6. ****(ADVANCED TOPIC QUESTION) **(Non-homogeneous ODEs: Method of variation of parameters)

Kreyszig 9th Edition Problem set 2.10, problems 1, 2, 3, 4, 5, 6.

Remember to reduce to the form - especially important for number 3.

Solve the given nonhomogeneous ODE by variation of parameters or undetermined coefficients. Give a general solution. (Show the details of your work.)

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