## Transcribed Text

l. Solve the following equations. In part b), express the general solution in terms of
appropriate parameters.
a) cos3
(x) tanh(y)dy - sin(x)dx = 0, y(O) = O;
b) d��t) + 2
d
2
d
��t
) + x(t) = 0.2.
a) Compute .C(eLet .C denote
2
the Laplace transform.
t sin(1rt)).
b) Prove that
.C (it f(u)du) = }.c(f).
c) Compute the Laplace transform of
J(t) = { s
�
n t,
t )
d) Compute .C(t sin(t)).
0 � t < 21r;
t � 21r. 3. Consider the family F of circles given by
F : x
2 + (y - c )
2
= c
2
, c E R
a) Write down an ODE y' = F(x, y) which defines the direction field of the trajectories in F. Draw a sketch.
b) Write down an ODE which defines a direction field perpendicular to the one you
found in part�- That is, find a direction field whose slope at (x, y) in the phase
plane is orthogonal to the slope given by F(x, y). Draw a sketch. [Hint. Use the
fact that if y1 and y2 are orthogonal curves, then at the point of intersection:
dy1 . dy2
= -
1.] dx dx
c) Find the curve through (1, 1) which meets every circle in the family F at an
angle of 90°
. Draw a sketch. [ Hint. Recall that the angle of intersection between
two curves is defined as the angle between their tangent lines at the point of
intersection.]4. Consider the linear 2nd order ODE:
(b) y" - 2y' + 2y = R(t).
Solve (b) in the following two cases, with R(t) given below:
a) R(t) = O; (give the general solution)
b) R(t) = et sec(t). (give a particular solution)5. (10 pts) Solve the IVP:
y" + 2y' + 2y = 821r (t), y(0) = 0, y'(0) = 0, t > 0.
(Here, 821r is the Dirac distribution at 21r.) Where is your solution discontinuous?
Where is it not differentiable? 6.(20 pts) Consider the Legendre equation
(q) (1 - x
2
)y" - 2xy' + 2y = 0.
a) Find two linearly independent solutions about x = 0. (You must solve completely any relevant recurrence relations.)
b) Compute the radius of convergence for each fundamental solution in part a).
c) Show that x = 1 is regular singular point of (q).
d) Assuming that y = (x-1)5 I:� an (x-lt (withs E JR) is a solution near x = l,
solve the indicial equation satisfied by s. Find one fundamental solution of (q)
near x = l.7. Solve the following system:
{ x� (t) = -llx1 (t) + 6x2(t)
x;(t) = -16x1(t) + 9x2(t)
with initial conditions x1 (0) = -2, x2(0) = -2.

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