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1st order linear differential equation:
P, Q – continuous.
dy
 P( x) y  Q( x)
dx
Algorithm:
Find I(x), s.t.
I ( x)  ( y  P( x) y )  ( I ( x)  y )  I ( x)  Q( x)
by solving differential equation for I(x):
I  y  I  P  y  ( I  y )  I  y  I   y 
dI
 I ( x)  P( x) 
dx
P ( x ) dx

I ( x)  e
then integrate both sides of the equation:
( I ( x)  y )  I ( x)  Q( x)  I ( x)  y   I ( x)  Q( x)dx  C
Simplify the expression, if possible.
2nd order linear differential equation:
d2y
dy
P( x) 2  Q( x)  R ( x) y  G ( x)
dx
dx
P, Q, R, G – continuous.
If G(x)0 equation is homogeneous, otherwise – nonhomogeneous.
2nd order linear homogeneous differential equation –
d2y
dy
P( x) 2  Q( x)  R( x) y  0
dx
dx
1). If y1 and y2 are solutions, then y=c1y1+c2y2 (linear combination) is also a solution.
2). If y1 and y2 are linearly independent solutions and P0, then the general solution is
y=c1y1+c2y2.
2nd order linear homogeneous differential equation
with constant coefficients:
ay  by  cy  0.
Find r, s.t. y(x)=erx is a solution (substitute into the equation):


e rx ar 2  br  c  0, e rx  0  ar 2  br  c  0
characteristic equation
 b  b 2  4ac
r
2a
Case I.
b 2  4ac  0
 b  b 2  4ac
Two unequal real roots r1 
2a
and
 b  b 2  4ac
r2 
2a
r1 x
r2 x
y

e
and
y

e
.
Therefore, 2 linearly independent solutions
1
2
General solution:
y  C1e r1x  C2e r2 x .
Case II. b 2  4ac  0
One real root r1  r2  r  
b
2a
Two linearly independent solutions are y1  e
and y2  xerx .
y  C1e rx  C2 xerx .
General solution:
Case III.
rx
b 2  4ac  0 (i.e. 4ac  b 2  0) 
b 2  4ac  i 4ac  b 2 , where i   1.
Two complex roots
r1    i and r2    i ,
b
4ac  b 2
where    ,  
2a
2a
Two linearly independent solutions
General solution:
y1  ex cos x and y2  ex sin x.
y  ex C1 cos x  C2 sin x .
Problems
1. Area between curves:
Set up the area between the following curves:
a) y  2 x  1, y  9  x 2
2
b) x  y  1, x  2 y
Sec. 6.1 5-26
2. Volumes by washer method (Sec. 6.2) and by cylindrical shells method (Sec 6.3):
Find the volume of the solid obtained by rotating the region bounded by y  x and y  x 2
about x=-1.
3. Arclength (Sec. 8.1):
Find the arclength of the curve
y  ln(cos x), 0  x 

3
.
4. Approximate integration (Sec. 7.7). Rn, Ln, Mn, Tn, Sn, formulae and error approximation:
How large should we take n for Trapezoid / Midpoint / Simpson’s Rules in order to
2
guarantee that the error of each method for
dx
1 x would be within 0.001?
Write down the expressions for R5, L5, M5, T5, S5.
5. L’Hospital Rule (Sec. 4.4):
(ln x) 2
Find lim
. Justify every time you apply L’Hospital rule!
x 
x
6. Improper integral (Sec. 7.8): type I, type II.
7. Differential equations:
1st order separable equation (Sec. 9.3): xy  y  2.
1st order linear equation (Sec. 9.6): x 2 y  xy  1.
2nd order linear equation (Sec. 17.1): 3 y  y  y  0, y  2 y  y  0, y  6 y  13 y  0.
Initial Value Problem and Boundary Value Problem.
8. Modeling: mixing problem, fish growth problem, population growth:
The population of the world was 5.28 billion in 1990 and 6.07 billion in 2000. Assuming
that the population growth rate is proportional to the size of population, formulate and
solve the corresponding differential equation. Predict world population in 2020. When will
the world population exceed 10 billion?
9. Recall: graphs and derivatives of elementary functions, integration techniques.
Product rule
Calculus
Chain rule
Implicit f-n
(about limits)
FTC
Differentiation
Quotient rule
Differential equations
Elementary f-ns:
Inverse processes
Integration
Optimization
Exact evaluation
Approximation
Applications
Polynomial
Rational
By parts
Substitution Riemann’s Other
(follows from (follows from sums
(Taylor’s)
product rule) chain rule)
under
Tn
R
n
Techniques of integration:
curve
S
Ln
n
Trigonometric integration
Algebraic
Power
Exponential
Logarithmic
Trigonometric
Hyperbolic/Inverse
Trigonometric substitution
Separable
between
curves
Arc Volume
length
washer
method
cylindrical
shells
Integral
Rational functions
Definite
number!
Differential equations
1st order
Mn
Area
Indefinite
function!
Initial Value problem
Boundary Value problem
Convergent
number!
2nd order linear
Linear Homogeneouos Non-homogeneouos
constant coefficients
Improper (Type I, II)
Mean Value Th
Intermediate Value Th
Extreme Value Th
Divergent
!