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MTH205: Practice Problems Exam 2 / Fall 07
Q1. Find an interval around x  0 for which the initial value problem
1
( x  1) y ' '
y ' y  sin( x) , y (0)  1 , y ' (0)  0
4  x2
has a unique solution.
Q2. Consider the differential equation
y  4 y  e 2 x
a) Find the general solution y c of the corresponding homogeneous equation.
b) Find a particular solution y p by the method of undetermined coefficients.
c) Find the general solution.
d) Determine the solution subject to the initial conditions:
y (0)  2 , y (0)  1
Q3. Solve the same problem in Q.2 by the method of variation of parameters.
Q4. Find the general solution of
x 2 y  4 xy  6 y  x3
Q5. Given that y1  e x is a solution of the d.e. ( x  1) y  ( x  2) y  y  0 , find the
general solution.
y1  1 is a solution of the d.e. y  y  0
order method to find the general solution of y   y   1 .
Q6. Given that
, use the reduction of
Q7. Set up the appropriate form of the particular solution yp , but do not determine
the values of the coefficients: y ( 4)  9 y  ( x 2  1) sin( 3x) .
Q8. Use the method of undetermined coefficients to find the solution of the I.V.P.
y  4 y  2 sin( 2t ) ,
y(0)  1, y(0)  0
Q9. Given that { y1  x 2 , y2  x 2 ln x } is a fundamental set of solutions of the
corresponding homogeneous equation of the differential equation
x 2 y  3xy  4 y  x 2 ln x
Use the variational method to find the general solution of the d.e.
Q10. A 16 – pound weight stretches a spring 2 feet. Initially the weight starts from
rest 2 feet below the equilibrium position. Determine the differential equation and the
initial conditions of the motion, if the surrounding medium offers a damping force
numerically equal to the instantaneous velocity and the weight is driven by an
external force equal to f (t )  5 cos( 2t ) . Describe the motion if there is no damping
force.
Q11. Find the transient and steady periodical solution of
y  2 y  2 y  20 cos(2t ) , y(0)  y(0)  0
Sketch the first 2 periods.
Q12. A mass weighing 64 pounds stretches a spring 0.32 foot. The mass is initially
released from a point 8 inches above the equilibrium position with downward velocity
of 5 ft/sec.
a) Find the equation of the motion.
b) Find the amplitude, natural frequency, period and phase angle of the motion.
c) At what time does the mass pass through the equilibrium position heading
downward for the second time?
d) At what times the mass attain its extreme displacements above the equilibrium
position?