18.034 FINAL EXAMINATION
May 22, 2007
Formulas of the Laplace transform (1) L[f (t)](s) = F (s) = f (t)e-st dt 1 (2) L[1] = . s 1 at (3) L[e ] = .
s - a
n! (4) L[tn ] = n+1
. s s
(5) L[cos(t)] = 2 . s + 2 (6) L[sin(t)] = 2 . s + 2 e-as (7) L
18.034 Midterm #1, Sample TF Questions The exam is Wed. 03/07/07, 1:00-1:55pm,
1. Every solution y = y(x) of the DE y + 2y = e-2x tends to zero as x . 2. The solution of the initial value problem y - y = 1 + 3 sin x, remains bounded as x . 3. Eve
18.034 Midterm #2
Name:
1. (a) (10 points) Find a linear differential equation with constant coefcients that has solutions t,
et and tet .
(b) (10 points) Prove or disprove that t4 and t6 can be solutions of one and the same linear ho
mogeneous differenti
18.034 Solutions to Problemset 4
Spring 2009
1. (a) u = u v + u1 v , u = u v + 2u v + u1 v ,
1
1
1
u = u v + 3u v + 3u v + u1 v .
1
1
1
(b) The equation for v reduces to (2 x)v + (3 x)v = 0, so that
v = c1 xex + c2 x + c3 . Hence, u = c1 x + c2 xex + c3
Problem set 9, Solution keys
1. Birkho-Rota pp. 135-136, Theorem 1 and Example 3.
Folia of Descartes are in gure 5.2.
2. (a) Suppose not. This means at least one of the inequalities f > 0, f < 0, g > 0, g < 0 holds
at (x1 , y1 ). Without loss of generali
LECTURE 8. UNIQUENESS AND THE WRONSKIAN.
Differential inequality and uniqueness. We prove the uniqueness theorem for linear secondorder differential equations with variable coefcients.
Theorem 8.1 (Uniqueness Theorem). If p(t) and q(t) are continuous on a
LECTURE 7. MECHANICAL OSCILLATION
The spring-mass system and the electric circuit. If a mass m is attached to one end of a suspended spring, it will produce an elongation, say y0 , which, according to Hookes law, is proportional to the force of gravity
ky
LECTURE 3. FIRST-ORDER LINEAR EQUATIONS
First-order linear differential equations. We will give a systematic method of solving rst-order
differential equations (of normal form)
y 0 + p(x)y = f (x)
(3.1)
on a given interval I, where p, f are continuous fun
UNIT I: FIRST-ORDER DIFFERENTIAL EQUATIONS
We set forth fundamental principles in the analysis of differential equations. We illustrate the
use of integration to nd the solutions of rst-order linear differential equations and special classes
of rst-order
LECTURE 5. LINEAR FRACTIONAL EQUATIONS AND SUBSTITUTION
Homogeneous equations. A function f (x, y) is said homogeneous of degree m if
f ( x, y) =
m
f (x, y),
> 0.
If P and Q are homogeneous of the same degree then P/Q is homogeneous of degree zero. Indeed
LECTURE 4. SEPARABLE EQUATIONS
Separable equations. Separable equations are differential equations of the form
dy
f (x)
=
.
dx
g(y)
(4.1)
For example, x + yy 0 = 0 and y 0 = y 2
differential form as
1. A separable equation (4.1) can be written in the
f (x
LECTURE 14. STABILITY
The notion of stability. Roughly speaking, a system is called stable if its long-term behavior does
not depend on signicantly the initial conditions.
An important result of mechanics is that a system of masses attached in (damped or
LECTURE 9. SEPARATION AND COMPARISON THEOREMS
Many references encourage the impression that computing the Wronskian of two functions is
a good way to determine whether or not they are linearly independent. But, two functions are
linearly dependent if one
LECTURE 13. INHOMOGENEOUS EQUATIONS
We discuss various techniques for solving inhomogeneous linear differential equations.
Variation of parameters: the Lagrange procedure. Let us consider the linear second-order differential operator
(13.1)
Ly = y 00 + p(
UNIT III: HIGHER-ORDER LINEAR EQUATIONS
We give a comprehensive development of the theory of linear differential equations with constant coefcients. We use the operator calculus to deduce the existence and uniqueness. We
presents techniques for nding a co
LECTURE 12. SOLUTION BASES
We present the results pertaining to the linear differential equation
L0 y = y (n) + p1 y (n
(12.1)
1)
+ + pn
1y
0
+ pn y
with constant coefcients. Some results we establish apply to equations with variable coefcients.
Let p( )
Problem set 7, Solution keys
1. (a)
(b) Y (s) =
P1 (s)
P2 (s) F (s)
y =wf
= W (s)F (s).
(c) Because W (s) 0 as s .
If P1 = P2 then W (s) = 1 and w(t) = (t). So, y = f .
2. (a)
(b) Once you have a cycloid:
E
T = 2 W H is calculated by mesuring the distan
18.034 Solutions to Problemset 5
Spring 2009
1. (a) Since fn f uniformly on [a, b], for > 0 given, there exists
N Z+ such that |fn (t) f (t)| < /(b a) for t [a, b] whenever
n N . For n N ,
b
b
b
fn (t) dt
f (t) dt
|fn (t) f (t)| dt <
a
a
a
(b) fn (t)
18.034 Midterm #1
Name:
Part I: TF Questions. Answer for each of the following statements if it is true or fale. Simply say T (if you believe it is true) or F (if you suspect it is false); you don't need to justify your answers. Each question count
18.034 Midterm #2
Name:
Part I: TF Questions. Answer for each of the following statements if it is true or fale. Simply say T (if you believe it is true) or F (if you suspect it is false); you don't need to justify your answers. Each question count
18.034 Midterm #2, Sample TF Questions The exam is Wed. 04/18/07, 1:00-1:55pm,
1. Consider the differential equation y n + a1 y n-1 + an y = e-x . If a1 , a2 , . . . , an are all positive, then every solution tend to zero as x . 2. Let 1 , 2 , 3
18.034 Practice Final Exams
The final exam will be held on Tuesday, May 22, 9:00AM12:00NOON. The final exam will be closed notes, closed book, calculators will not be permitted. A short list of Laplace trans forms will be provided. The following set
18.034 Midterm #1
Name:
1. (20 points) Solve the initial value problem
y y t = 0,
y (1) = 2,
1
y (1) = 1.
18.034 Midterm #1
Name:
2. Consider the differential equation y = y (5 y )(y 4)2 .
(a) (7 points) Determine the critical points (stationary solutions
18.034 Solutions to Problemset 1
Spring 2009
1. (b) a = 1 or a = 4
2. (a) y (/6) = e2
(b) Same as part (a)
(c) Because
x = 0.
dx
x
is not integrable on any interval containing the point
3. (a) y is increasing because y = y 2 + 1 > 0. The formula is obt
18.034 Practice Midterm #1
1. Solve the initial value problem
y 3y 2 = 0,
y (0) = 2, y (0) = 4.
Determine the interval in which the solution exists.
2. Consider the differential equation y = (1 y )(y 2)3 .
(a) Sketch the graph of f (y ) = (1 y )(y 2)3 .
(
18.034 Practice Midterm #3
Notation. = d/dt.
1. (a) If f E and F (s) = L[f (t)], show that lims F (s) = 0.
s+1
(b) Find the inverse Laplace transform of F (s) = log
.
s1
2. (a) Sketch the graph of f (t) = (1/5)(h(t 5)(t 5) h(t 10)(t 10), where h(t) is the
18.034 Practice Midterm #2
Notation. = d/dt.
1. (a) Find numbers a and b so that the differential equation t2 y + aty + by = 0 has solutions t2
and t3 on the interval t (0, ).
(b) Find a differential equation that has solutions (1 t)2 and (1 t)3 on the in
18.034 Solutions to Problemset 3
Spring 2009
1. (b) y1 (0) =
(c) y2 (0) =
1
+0 0
1
+0
(d) lim y2 (t) =
0
+ as 0 .
21 .
1
t cos t.
2
2. Birkho-Rota, pp. 28, Theorem 5.
3. (a) c1 cos x + c2 sin x
x
x
(b) c1 x + c2 e2x
4. (c)
1
n(n + 1)
+
(n +
18.034 Midterm #3
Name:
1. (a) (15 points) If f E and f is continuous, show that lims sF (s) = f (0).
(b) (5 points) Can F (s) = 1 be the Laplace transform of a function f E ?
1
18.034 Midterm #3
Name:
2. (a) (10 points) Show that the solution of the init
18.034 Practice Final
Notation. = d/dt.
1. (a) Is the differential form x2 y 3 dx + x(1 + y 2 )dy exact?
(b) Find a function (x, y ) so that (x, y )(x2 y 3 dx + x(1 + y 2 )dy ) becomes exact.
(c) Solve the differential equation
dy
x2 y 3
=
.
dx
x(1 + y 2