Math 22B Solutions Homework 2 Spring 2008
Section 2.1 16. y + 2 y = t
cos t , t2
with y() = 0 and t = 0
R
2 dt t
Solution Let (t) = e equation by (t), we get:
= et . If we multiply both sides of the given 2 = cos t t
2
t2 (y) +
(t2 y) = cos t
MAT 22B 001
3.2. Solutions of Linear Equations
9 APR 2014
3.2 Solutions of Linear Equations
A. Existence and Uniqueness
Thoerem 3.2.1 (EUT)
Let the functions p, q, g be continuous on an open interval
1 < t < 2, containing the point t0; then, on the interv
MAT 22B 001
1.2. Solutions of Dierential Equations
2 APR 2014
1.2 Solutions of Dierential Equations
A. Solutions
We want to understand the unknown function in the given dierential
equation; in particular, we shall address the following questions.
Solutio
MAT 22B 001
2.1. Linear Equations
4 APR 2014
2.1 Linear Equations
A. First-Order Linear Equations
A rst-order linear ODE is an ODE that can be written as
y + p(x) y = q(x).
When p(x), q(x) are constant functions, the equation specializes to
y ay = b or eq
MAT 22B 001
2.2. Separable Equations
7 APR 2014
2.2 Separable Equations
A. First-Order Separable Equations
A rst-order separable ODE is an ODE that can be written as
h(y)
dy
= g(x).
dx
When g(x) = 1 and h(y) = (ay + b)1, the equation specializes to
1
y =
MAT 22B 001
3.1. Linear Homogeneous Equations I
24 APR 2014
3.1 Linear Homogeneous Equations I
A. Second Order Linear Equations
A second-order linear ODE is an ODE that can be written as
y + p(t) y + q(t) y = g(t)
We say its homogeneous if g(t) is the con
MAT 22B 001
2.8. Existence and Uniquness
14 APR 2014
2.8 Existence and Uniquness
A. Restatement of EUT
Whenever we consider an initial value problem y = f (t, y), y(t0) = y0,
we can apply a simple preliminary change of variables to convert the
problem int
MAT 22B 001
2.3. Modeling with First-Order Equations
9 APR 2014
2.3 Modeling with First-Order Equations
A. One-Compartment Mixing Models
A one-compartment mixing model concerns the amount of a
chemical substance present in the solution in a single contain
MAT 22B 001
1.1. Utility of Dierential Equations
31 MAR 2014
1.1 Utility of Dierential Equations
A. Dierential Equations
A dierential equation is an equation involving variables, an unknown function in these variables, and its derivatives.
The unknown fun
MAT 22B 001
2.4. Linear vs. Non-Linear
14 APR 2014
2.4 Linear vs. Non-Linear
A. Existence and Uniqueness of Solutions
The existence and uniqueness theorem (EUT) stipulates that
there exists a unique solution of an initial value problem, provided that
some
3. 6 \[a CLWM cg iDCKWKe/g.
%ll_gkj/ Zn 3 48-21;
® 3k mm as
(ZIBYJver $(V47CF-2) Co 97 r: I Z
:3 3% = Q at + C; ezt < W-z WWW
, t
5 U1 Qt +Uzzéi+ (e/f "IL 62'
,JW
1 _ V; .gt 3
2 Uker £12261 W H S W 2
. u . t (if Q/t+uz/ eat :0
= Wen me + MZQéH alike?
:
MAT 22B 001
1.3. HW Solutions
1.3
15 APR 2014
HW Solutions
#2. Second-order non-linear ODE (the coecient in front of y makes it non-linear).
#4. First-order non-linear ODE (the presence of y 2 term makes it non-linear).
#6. Third-order linear ODE (the coe
1.2. HW Solutions
MAT 22B 001
1.2
15 APR 2014
HW Solutions
#3. y = ay + b with a > 0, b > 0.
(a) (ay + b)1 dy = dt ln |ay+b| = t + c1 ln | ay + b| = at + c2
a
| ay + b| = eat+c2 = ec2 eat ay + b = c3 eat y = Ceat + (b/a) with C = 0;
however, the equilibr
MAT 22B 001
1.1. HW Solutions
1.1
14 APR 2014
HW Solutions
#7. There is no unique solution to this problem; we look for an equation of the form
y = ay + b. For this equation to have an equilibrium y = 3, we need b = 3a, so
we have y = ay 3a. If we take a
7 1 Msxr 005;,
Wekowe gunned :, 14: HM)
We, wow 8+ ony SYServxS ; \ZM. : H W va/Jg/c
\2 SM ' _ gr;
I I szrvxcf LQT+ ,
1;:Flbcljwrxn3t3 1) IX pvourw
E : t \de_
/ l \
DC" = H DC; DC. at) KW WI so WE
Ear m:de \ichéL Pmbbamg, W SYwQY WWW» 14
Pr Sjsfm TS sa to
2.5. Autonomous Equations
MAT 22B 001
14 APR 2014
2.5 Autonomous Equations
A. Equilibria of Autonomous Equations
A constant solution y(x) = y to an ODE is called an equilibrium.
We study the equilibria of autonomous equations of the form
y = g(y).
Observa
MAT 22B 001
1.3. Classication of Dierential Equations
4 APR 2014
1.3 Classication of Dierential Equations
A. Ordinary and Partial
An ordinary dierential equation is a dierential equation with
the unknown function in a single independent variable, and
it
Homework 3: Math 22B - Tavernetti, Fall 2016
Due Wednesday, Oct 12 in class
(20 points)
Instructions : Solve all problems. Print out your solutions when computer results are asked for
(do not include the dfield8.m code), work neatly, label your plots, sho