PRACTICE PROBLEMS FOR FINAL
(1) Consider the equation
y 0 − 3x2 y = 0
(a) Solve it by using the method of integrating factors.
(b) Find the power series satisfying the equation, and check that it is the
same answer as in a). You may use that
ex =
∞
X
xn
n!
n=0
(2) Find the explicit solution of
cos x
y0 =
,
y(0) = 0
2y + 1
What is the maximum interval of existence of the solution?
(3) Consider the ODE
ex y + aesin y
cos y − bex
Find some real constants a, b so that b 6= 0 and the equation is exact.
For those constants a and b, find an implicit solution.
y0 =
(4) What can we say about the maximum interval of existence of the solution
to the equation
p
(t2 − 1)y 0 + (tan t)y = 2 − t2 ,
y(0) = 2?
(5) Find the solution of the DE
y 00 + 5y 0 + 4y = 0,
y(0) = 1,
y 0 (0) = 0
You may use any method you like!
(6) Consider the ODE
x2 y 00 − x(x + 2)y 0 + (x + 2)y = 0,
x>0
Given that y1 = x is a solution, find a second solution y2 so that W (y1 , y2 )(1) =
2e.
(7) What is the proper form of a particular solution to
y 00 + 7y 0 = x2 − ex cos x
in the method of undetermined coefficients? You don’t need to actually
solve the equation.
(8) Solve the equation
y 00 + 4y 0 + 4y =
p
1 + t2
Your answer may include integrals that cannot be evaluated exactly!
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PRACTICE PROBLEMS FOR FINAL
(9) Find the explicit solution of the equation
y 0 − y = 2tδ(t − 1) + tu2 (t),
y(0) = 0
(10) By using the Laplace transform, solve the DEs
y 00 + 6y 0 + 10y = cos t,
y(0) = 1,
y 0 (0) = 0
You will not get any credit for solving it any other way!
(11) Let

 1 0 ≤ t < 2,
t 2 ≤ t < 6,
f (t) =
 2
t
6 ≤ t.
P
(a) Write f in terms of elementary step functions, f (t) = uci (t)gi (t−ci )
(b) Compute the Laplace transform of f .
(12) Solve the DE
y 000 − y = sin t2 ,
y(0) = y 0 (0) = y 00 (0) = 0
Your answer may be given as a definite integral, but you must explicitly
identify the correct integral bounds and the function to be integrated.
(13) Consider the DE
y 00 − x2 y = 0,
y(0) = 0,
y 0 (0) = 1
i) Is 0 an ordinary or singular point? What do we know of the existence
of a power series solution
∞
X
an xn ?
y(x) =
n=0
ii) If a power series solution exists, compute a0 , a1 , a2 , a3 and a4 .
iii) If a power series solution exists, write down the recurrence relation.
(14) Consider the ODE
(a)
(b)
(c)
(d)
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x3 y 00 − x2 y 0 + (ex − 1)y = 0, x > 0
2
Is 0 a regular singular point? Justify your answer!
If the answer to the question above is yes, determine the roots of the
indicial equation.
P
Are we guaranteed to have two Frobenius solutions of the form xr an xn ?
Justify your answer!
If the answer to the previous question is yes, write down the general
form of the two solutions, and compute a1 in terms of a0 .
PRACTICE PROBLEMS FOR FINAL
Helpful Laplace transforms
L{eat cos bt} =
s−a
(s − a)2 + b2
L{eat sin bt} =
b
(s − a)2 + b2
L{tn eat } =
n!
(s − a)n+1
L{t sin at} =
2as
(s2 + a2 )2
L{δ(t − c)} = e−cs
L{uc (t)f (t − c)} = e−cs L{f }
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