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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 P0, 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 ex cos x and y2 ex sin x. y ex 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 !