HW 3 Mathematics 503, Mathematical Modeling, CSUF June 2 2007

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HW 3 Mathematics 503, Mathematical Modeling, CSUF June 2 2007

Nasser M. Abbasi

June 15, 2014

1 Problem 1 (section 1.3,#1, page 40)
2 Problem 2 (section 1.3,#1, page 40)

1 Problem 1 (section 1.3,#1, page 40)

problem: Find the general solution of the following diﬀerential equations

(m) $u ′′+ ω2u= sinβ t$ , $ω ⁄= β$

solution:

We start by assuming $ω$ is real, hence $2 ω$ must be positive.

Now, the general solution is

Where $u1(t),u2(t)$ are the 2 independent solutions of the homogeneous diﬀerential equation

u′′+ ω2u = 0

and $up(t)$ is the particular solution.

To ﬁnd $u1(t)$ and $u2(t)$ , we assume the homogeneous solution is $mt uh(t)= Ae$ for some constants $A,m$ and substitute this assumed solution in the ODE. We obtain the characteristic equation $Am2emt+ ω2Aemt = 0→ m2+ ω2 = 0 → m2 = − ω2$ or $m = ±iω$ , hence $u1(t)= A1eiωt$ and $u2(t)= A2e−iωt$ .

Since the homogeneous solution is a linear combination of all the independent solutions, we take the sum and the diﬀerence of these solutions, and using Euler relation which converts the complex exponential to the trigonometric $sin$ and $cos$ functions we obtain

and we now write

uh(t) = c1cosωt+ c2sin ωt

Now to obtain $up(t)$ , we use the method of undetermined coeﬃcients. Assume

up(t)= acosβt+ bsinβ t

and plug into the original ODE, we obtain

Hence by comparing coeﬃcients we obtain

Since $ω ⁄= β$ (given), then $2 2 ω − β ⁄= 0$ , hence this must mean the following

Therefor, the particular solution now can be written as

Hence the general solution, which is $y (t)= yh(t)+ yp(t)$ is given by

|--------------------------------| | sinβt | | y(t) = c1cosωt+ c2sinωt+ ω2−β2- | ----------------------------------

Where $c1$ and $c2$ are constants that can be found from the initial conditions.

2 Problem 2 (section 1.3,#1, page 40)

problem: Find the general solution of the following diﬀerential equations

(n) $′′ 2 u + ω u = cosωt$

solution:

First, let the forcing function be called $f (t)$ , hence $f (t)= cos ωt$ in this example.

From (m) we found the homogeneous solution to be $uh(t) = c1u1(t)+ c2u2(t)$ where

u1(t)= cosωt

and

u2 (t)= sinωt

Now to ﬁnd the particular solution we can not use the method of undetermined coeﬃcients since the forcing frequency is the same as the undamped natural frequency of the system $ω$ and this will lead to the denominator going to zero. Hence use the method of variation of parameters which is a general method to ﬁnd particular solution which will work with this case.