1 st order linear differential equation: p, q – continuous. algorithm: find i(x), s.t. by solving...
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1st order linear differential equation:
P, Q – continuous.)()( xQyxP
dx
dy
Algorithm:
Find I(x), s.t.
)()())(())(()( xQxIyxIyxPyxI
)()()( xPxIdx
dIyIyIyIyPIyI
by solving differential equation for I(x):
dxxP
exI)(
)(
CdxxQxIyxIxQxIyxI )()()()()())((
then integrate both sides of the equation:
Simplify the expression, if possible.
2nd order linear differential equation:
P, Q, R, G – continuous.
)()()()(2
2
xGyxRdx
dyxQ
dx
ydxP
If G(x)0 equation is homogeneous, otherwise – nonhomogeneous.
2nd order linear homogeneous differential equation –
0)()()(2
2
yxRdx
dyxQ
dx
ydxP
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 P0, then the general solution is y=c1y1+c2y2.
2nd order linear homogeneous differential equation with constant coefficients:
.0 cyybya
Find r, s.t. y(x)=erx is a solution (substitute into the equation):
00,0 22 cbrarecbrare rxrx
characteristic equation
a
acbbr
2
42
Case I.
Two unequal real roots and
Therefore, 2 linearly independent solutions
General solution:
042 acb
a
acbbr
2
42
1
a
acbbr
2
42
2
.2121
xrxr eyandey
.2121
xrxr eCeCy
Case II.
One real root
Two linearly independent solutions are
General solution:
042 acb
a
brrr
221
.21rxrx xeyandey
.21rxrx xeCeCy
Case III.
Two complex roots
Two linearly independent solutions
General solution:
.1,44
)04..(04
22
22
iwherebaciacb
baceiacb
a
bac
a
bwhere
irandir
2
4,
2
,
2
21
.sincos 21 xCxCey x
.sincos 21 xeyandxey xx
Problems
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
about x=-1.
2xyandxy
1. Area between curves:
Set up the area between the following curves:
a)
b)
Sec. 6.1 5-26
29,12 xyxy 22,1 yxyx
3. Arclength (Sec. 8.1):
Find the arclength of the curve .3
0),ln(cos
xxy
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
guarantee that the error of each method for would be within 0.001?
Write down the expressions for R5, L5, M5, T5, S5.
2
1 x
dx
5. L’Hospital Rule (Sec. 4.4):
Find Justify every time you apply L’Hospital rule!.)(ln
lim2
x
xx
6. Improper integral (Sec. 7.8): type I, type II.
7. Differential equations:
1st order separable equation (Sec. 9.3):
1st order linear equation (Sec. 9.6):
2nd order linear equation (Sec. 17.1):
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.
.2 yyx
.12 xyyx
.0136,02,03 yyyyyyyyy
Calculus (about limits)
Differentiation IntegrationInverse processes
FTC
Product rule
Chain rule
Implicit f-n
Quotient rule
Exact evaluation Approximation Applications
By parts(follows from product rule)
Substitution(follows from chain rule)
Techniques of integration:
Trigonometric integration
Trigonometric substitution
Rational functions
Riemann’s sums
Other(Taylor’s)
Rn
Ln
Mn
Tn
Sn
Area VolumeArclength
under curve
betweencurves
washermethod
cylindricalshells
Integral
Definitenumber!
Indefinitefunction!
Improper (Type I, II)
Convergentnumber!
Divergent!
OptimizationDifferential equations
Differential equations
1st order 2nd order linear
Separable Linear Homogeneouosconstant coefficients
Non-homogeneouos
Elementary f-ns:
Polynomial
Rational
Algebraic
Power
Exponential
Logarithmic
Trigonometric
Hyperbolic/Inverse
Mean Value Th
Intermediate Value Th
Extreme Value Th
Initial Value problemBoundary Value problem