Page 317 - A Course in Linear Algebra with Applications

8.3: Applications to Systems of Linear Differential Equations 301

3. By triangularizing the coefficient matrix solve the system
of differential equations

(y'i = 5yi + 3y 2
-
\ 2/2 = 3yi - V2
Then find a solution satisfying the initial conditions yi(0) = 0,
y 2(0) = 2.
4. Solve the second order linear system

y'( = 2j/i + y 2 + y[ + y' 2
2
y'i = ~ %i + 2/2 + 5yi - y' 2

5. Given a system of n (homogeneous) linear differential equa-
tions of order k, how would you convert this to a system of
first order equations? How many equations will there be in
the first order system?
6. Describe a general method for solving a system of second
order linear differential equations of the form Y" = AY, where
A is diagonalizable.
7. Solve the systems of differential equations

y'i = y\ - 2/2 (h) [y'i = - 4y- 2
y'{ = 33/1 + 5y 2 { M y'{ = Vl + 5y 2

[Note that the general solution of the differential equation
u" = o?u is u = cicosh(ax) + c 2sinh(ax)].
8. {The double pendulum) A string of length 21 is hung from
a rigid support. Two weights each of mass m are attached
to the midpoint and lower end of the string, which is then
allowed to execute small vibrations subject to gravity only.
Let y\ and y 2 denote the horizontal displacements of the two
weights from the equilibrium position at time t.
(a) (optional) By using Newton's Second Law of Motion,
show that yi and y 2 satisfy the differential equations