and do the first two
iterations.
Dr. Cleary
1) Give an example of a first-order differential equation which has more than one solution for an initial value problem. Are there any second-order differential equations that also have this property?
2) Set up the Picard iteration technique to solve the
initial value problem and do the first two
iterations.
3) On what interval is the solution of 
with initial condition 
guaranteed by our 1st
order existence theorem?
4) For what initial conditions can we guarrantee uniqueness of solution for the differential equation of question 3?
5) Is the following set of functions a basis for the
solutions to the second order equation: 
:
? If not, find such a basis.
6) Is the following a basis for the solutions to the
second order differential equation 
:
? If not, find such a basis.
7) Solve the differential equation 
.
8) Find all complex 4th roots of 8 - 6i.
9) How many of the solutions to 
have 
?
10) Compute the Wronskian of the set 
.
Is it a linearly independent set?
11) Find a basis for the solutions to 
12) Find the general solution to the differential
equation 
13) An underdamped mass-spring system is set into motion. If the mass starts at 10 cm above the rest position and has a downward push of 5 cm/sec, draw a sketch of the resulting motion.
14) Solve the differential equation :
15) Solve the differential equation 
subject
to the conditions that y(0) = 4 and y(10) = -1.
15) Prove the following or give an example to show that it is
false: For any complex number z,
.
16) Find another independent solution to the equation
given that we notice the ``obvious'' solution
of 
.
17) Find the general solution to 
18) Find the general solution to 
19) Sketch the output of the following Mathematica code:
Do[sol[i]= DSolve[{y''[x]-2 y'[x] + y[x] == 0, y[0]==10, y'[0] == i*10},
y[x],x] [[1,1,2]],{i,-4,4}]
Plot[Evaluate[Table[sol[n],{n,-4,4}]],{x,0,2}]
20) Solve the following differential equation: 
by
introducing a new variable 
and eliminating 
.