Math 2280 - Assignment 6

Dylan Zwick

Spring 2013

Section 3.7 - 1, 5, 10, 17, 19

Section 3.8 - 1, 3, 5, 8, 13

Section 4.1 - 1, 3, 13, 15, 22

1

Section 3.7 - Electrical Circuits

3.7.1 This problem deals with the RL circuit pictured below. It is a series circuit containing an inductor with an inductance of L henries,a resistor with a resistance of R ohms, and a source of electromotiveforce (emf), but no capacitor. In this case the equation governing oursystem is the first-order equation

LI’ + RI = E(t).

L

Suppose that L = 5H, R = 25Q, and the source E of emf is a batterysupplying 100V to the circuit. Suppose also that the switch has beenin position 1 for a long time, so that a steady current of 4A is flowingin the circuit. At time t = 0, the switch is thrown to position 2, sothatl(0)=4andE=Ofort>0.Findl(t).

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3.7.10 - This problem deals with an RC circuit pictured below, containinga resistor (R ohms), a capacitor (C farads), a switch, a source of emf,but no inductor. This system is governed by the linear first-orderdifferential equation

R+-Q=E(t).

for the charge Q = Q(t)Q’(t).

on the capacitor at time t. Note that 1(t) =

C

Suppose an emf of voltage E(t) = cos wt is applied to the RCcircuit at time t = 0 (with the switch closed), and Q (0) = 0. Substitute

Q8(t) = A cos wt + B sin wt in the differential equation to show thatthe steady periodic charge on the capacitor is

E0CQ8 (t) =

_________

cos (wt—

3)p \/l+w2R2C2

where 3 = tan’ (wRC).

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3.7.17 For the RLC circuit pictured below find the current 1(t) using thegiven values of R, L, C and V(t), and the given initial values.

L

=t ?r*Z L

—1± _cI -

e Yi4(1()

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Rzrrl6Q,L=2H,C=.02F;

/6E(t) = 100V; 1(0) = 0, Q(O) = 5.

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7

Section 3.8 - Endpoint Problems and Eigenvalues

3.8.1 For the eigenvalue problem

y”+)y=O; y’(O) =O,y(l)=O,

first determine whether ) = 0 is an eigenvalue; then find the positiveeigenvalues and associated eigenfunctions.

A

A A ()+8si1(K)

y ((a) A c y ‘() - A A f () (; ()

$0)

y1(o) 0

*1jVi1 To(,Or1 ( Awôikc (iivl 5al Ii4)) kY eJ7eIVaftt(’.

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9

3.8.3 Same instructions as Problem 3.8.1, but for the eigenvalue problem:

y”+y = 0; y(—7r) = O,y(ir) = 0.

y& 0J()tfifi4()

y -A c() o()

C

\/(T1 Aoi()t 1(d 0

y(- co5( cA ( ) - 81 ()

,z\V0) /H0

5 Z A c o () = o3ol4( FLI ;o A =0 07 r (J )

1 ( t) H efli1 (()) C)

5i I1Z4 A=o 4-h,i\/( SJi()

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Q)7vcu1 e1

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10 y( cs(’fLx)

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3.8.5 Same instructions as Problem 3.8.1, but for the eigenvalue problem:

y” + \y = 0; y(—2) = 0, y’(2) = 0.

t Ly(() A

-s

CdS

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If Ayy1

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/( Acar(x)Bc

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e ii AP/

z)=Icf

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k= [ (i11Tr1

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3.8.8 - Consider the eigenvalue problem

y”+Ay= 0; y(O) =0 y(l)= y’(l)(notatypo).;

all its eigenvalues are nonnegative.

(a) Show that ). = 0 is an eigenvalue with associated eigenfunction

Yo() = x.

(b) Show that the remaining eigenfunctions are given by yTh(x)sin/3x, where /3 is the nth positive root of the equation tan z =

z. Draw a sketch showing these roots. Deduce from this sketchthat (2n + 1)ir/2 when n is large.

Bo

y(i)A \/ ()

5 o e ct / e 1417 in ci /1( J /

y( x.A70

(05 ( ) t () y (i) ()/ (a) .4 0 ‘(1) ()yI i() ri(v) Co()

y ‘( I (°5( K) c (P o ,t }ke1/vtI rc1k)

12

More room, if necessary, for Problem 3.8.8.

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13

3.8.13 - Consider the eigenvalue problem

y”+2y’+AyO; y(O)=y(l)=O.

(a) Show that A 1 is not an eigenvalue.

(b) Show that there is no eigenvalue A such that A < 1.

(c) Show that the nth positive eigenvalue is A = n2ir2 + 1, withassociated eigenfunction y(X) = e sin (n7rx).

1’7rI roch- -f

y(H A eA O

y (i)

50 SO U ke 4riv(qf fG/cIi 04

ere ( f /1 c € ‘C7 /t(e

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y() A e

A=cy( Ae÷Oe

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Section 4.1 - First-Order Systems and Applications

4.1.1 - Transform the given differential equation into an equivalent systemof first-order differential equations.

x” + 3x’ + 7x = t2.

D xl 7

50/ -)1€ 5yfk i

r71: xrej

16

4.1.3 - Transform the given differential equation into an equivalent systemof first-order differential equations.

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(4) ‘I+ 6x — 3x’ + x = cos 3t.

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17

4.1.13 - Find the particular solution to the system of differential equationsbelow. Use a computer system or graphing calculator to construct adirection field and typical solution curves for the given system.

x’=—2y, y’=2x; x(O)=1,y(O)=O.

y yy y y () A t

y -?A//)2c(?)

.- Af(7cs(?)

/

18

More room, if necessary, for Problem 4.1.13.

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19

4.1.15 - Find the general solution to the system of differential equationsbelow. Use a computer system or graphing calculator to construct adirection field and typical solution curves for the given system.

y - / y’=-8x.

y 4YyO

y () A () / ()y () y ‘(i) 5/ (z )

&ffl era) 5 / !:

x () - si ( ) + cc ( )y If) A f (?)

20

More room, if necessary, for Problem 4.1.15.

Pirecio

21

4.1.22 (a) - Beginning with the general solution of the system from Problem 13, calculate x2 + y2 to show that the trajectories are circles.

(b) - Show similarly that the trajectories of the system from Problem15 are ellipses of the form 16x2 + y2 = C2.

X ( i -A j’ (Ze) 8 sy() A coC7)fI?)

X (] y () A Ifl (?) G5 (A o(?#)mi

A c () A 611 “1 (Z5) (0$2

A cc () I

D A L+ c[rc ( .

b) ejq/ 5 / i:

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