Exercises #13 Solving Differential Equations and First-Order Systems
1. Find the general solution for the following differential equations:
(a) 0107 yyy (b) 05314 yyy
(c) 034407 yyyy
2. Solve the following initial value problem: 024269 yyyy with initial
conditions 6)0( y , 13)0( y and 33)0( y .
3. Find solutions for the following first-order systems:
(a) XX
8213
4112
6185
dt
d (b)
yxy
yxx
2
133 with
2)0(
9)0(
y
x.
4. The real-valued 33 matrix A has eigenvalues and eigenvectors as described below:
Eigenvalue, –2 i53
Eigenvector T4,1,2 Tii 1,5,72
Find the general solution to the first-order system XX
Adt
d .
5. For the differential equation in Question #2 do the following:
(a) Rewrite the nth order differential equation as a first-order system in three unknown
functions: XX
Adt
d .
(b) Either find the eigenvalues of A or “guess” them from your work in #2.
(c) Either find the corresponding eigenvectors of A or “guess” them. If you are
“guessing”, check your guess using the matrix A.
(d) Give the general solution of the first-order system and of the function y .
6. The electrical circuit shown (called a parallel LRC circuit) contains a resistor with
resistance R, an inductor with inductance L and a capacitor with capacitance C. It is
possible to show that the current I through the inductor and the voltage drop V across
the capacitor satisfy the system of differential equations:
RC
V
C
I
dt
dV
L
V
dt
dI
Find the functions I and V if 5.0L H, 2.0C F and 5.2R and we are given
initial conditions 1)0( I A and 1)0( V V.
L
R
C
F J -���c±c,--w Ai -= I R-I� Lf -Ii
d--- -1 d-.
'3, -J..l
X-=0
[�]s J := �s �
-z._ ::: s
>..J- -= �: A- ai == 3 -lB
X-:::=. -�s
j - 0 � -
z_. =s
J\ 3 -=--\ ·, A +"L
-v - J.. s " - J.
u - J..s � v - �
'"<.. = s
� -13
3 -�-l
(-;] s·, Co
.J..] J.. s 1
I
3
-li-lo
-11
@)
lo r,.i.f 0 0
lf 0 -'/.
-� 3
-=,-0 0 0
� 0
CloOO-Q_ S' ='3 " V -::: I
3·
lo
rr�-f- . I 0 1
'-f ---:) 0 I -0
<.o 0 0 0
�
� Cf]CloOO-R... s -=- I . V:i
(p 0
_J-
rriZ:t .;i..
l..( ---3 0 -fi
� 0 0 0
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