Math 24 Final Review Problems
August 26, 2015
This is a short list of the types of differential equations we’ve learned to solve
throughout this class. This is meant to be an example of the different types of
differential equations and techniques you have been shown and not a ‘mock’
exam. The problems are taken from the textbook and some of the solutions can
be found in the appropriate solutions pdf on the course website.
2.1.#17 Solve the initial value problem (linear, first order)
y 0 − 2y = e2t ,
y(0) = 2.
2.2.#5 Solve the differential equation (seperable, first order)
y 0 = (cos2 x)(cos2 2y).
2.5.#1 Find the equilibrium points, classify each one as stable/unstable, and draw
the phase lines for the following equation (autonomous, first order)
dy
−2(arctan y)
=
.
dt
1 + y2
2.5.#3 Determine whether the following equation is exact. If it is exact, find the
solution
(3x2 − 2xy + 2) = (6y 2 − x2 + 3)y 0 = 0.
2.8.#4 Using Picard’s method, compute the approximate solutions φn (t) for n =
1, 2, 3, 4 for the following equation
y 0 = −y − 1,
y(0) = 0.
3.5.#5 Find the general solution to
y 00 − 2y 0 − 3y = −3te−t .
(second order, linear, nonhomogeneous, repeated roots) note: also know
how to do complex roots case and non-repeated real root case.
1
3.8.#6 A mass of 5 kg stretches a spring 10 cm. The mass is acted on by an
external force of 10 sin(t/2) N (newtons) and moves in a medium that
imparts a viscous force of 2 N when the speed of the mass is 4 cm/s. If the
mass is set in motion from its equilibrium position with an initial velocity
of 3 cm/s, formulate the initial value problem describing the motion of
the mass.
7.5.#9 Find the general solution to the given system of equations. Draw a direction field and a few of the trajectories.
4 −3
0
x =
x.
8 −6
7.7.# 6 Find the fundamental matrix Φ(t) for the given system of equations satisfying Φ(0) = I
−1 −4
x0 =
x.
1 −1
7.9.#7 Find the general solution to the given system of equations
−1 1 1
t
x0 =
x+
,
t > 0.
4 1
2t−1 + 4
5.2.#6 Seek the power series solution to the following differential equation at the
point x0 = 0. Find the recurrence relation for the coefficients of the power
series. Compute the first four terms of each of the two solutions y1 and
y2 . Evaluate the wronskian W (y1 , y2 )(x0 ). If possible, find the general
term in each solution (this means solving the recurrence relation).
(2 + x2 )y 00 − xy 0 + 4y = 0,
2
x0 = 0.