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# MATH:340 Introduction to Ordinary Differential Equations Exam 1 Study Guide MATH340

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This 15 page Study Guide was uploaded by Raffasarru on Sunday October 2, 2016. The Study Guide belongs to MATH340 at Colorado State University taught by Peter A Muller in Fall 2016. Since its upload, it has received 46 views. For similar materials see Introduction to Ordinary Differential Equations in Math at Colorado State University.

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Date Created: 10/02/16

MATH:340 Introduction to Ordinary Differential Equations Exam #1 Study Guide Three Sections: Prerequisite Review, Ordinary Differential Equations, Matrix Algebra Highlighting: Used OFTEN. KNOW THIS. Used sometimes, good to know. 1. Prerequisite Review a. Partial Fraction Expansion: (explained by example) 7????−7 Example: ???? ???? =) ???? −8????+7 Rewrite as: 7????−7 = ???? + ???? ???? −8????+7 ????−1 ????−7 ( )( ) (????−1 ????−7 7????−7 Multiply both sides by ???? − 1 ???? − 7 : ????−1 ????−7 ???? −8????+7 = (????−1 ????−7 ???? ????−1 ????−7 ???? ????−1 + ????−7 ⇒ 7???? − 7 = ???? ???? − 7 + ???? ???? − 1 ) Set x’s so that either coefficient will be zero, solve, then do it for the other one. 1. Set ???? = 1: − 54 = ????(−6) so ???? = 9 2. Set ???? = 7: 48 = ????(6) so ???? = 8 7????−7 9 8 Rewrite: 2 = + ???? −8????+7 ????−1 ????−7 b. Polynomial Division: (explained by example) ???? +3????+6 ????+1 c. U-Substitution: Use when given ∫ ????(???? ???? )???? ???? ???????? then ∫ ????(???? ???? )???? ???? ???????? = ∫???? ???? ???????? where ???? = ????(????) 2 2ln ????) Example: ∫ ln ???? )???????? = ∫ ???? ???????? 1 1. Set ???? = ln(????) and ???????? = ???? 2. Substitute in∫ 2???? ???????? = 2 ????∫???????? ????2 2 3. Integrate: 2(2) + ???? = ???? + ???? 4. Substitute back in: ln (????) + ???? d. Integration by Parts: Use when ???? ???? has one part that can be integrated and one part that can be differentiated ∫ ???? ???? ????′(????)???????? = ???? ???? ???? ???? − ???? ????∫????(????)???????? so ∫ ???? ???????? = ???????? − ????∫???????? Trick for choosing u: LIATE 1. L: logarithmic functions 2. I: inverse trigonometric functions 3. A: algebraic expressions 4. T: trigonometric functions 5. E: exponential functions Example: ∫ ???? sin(????)???????? (infinite loop trick) ???? ???? 1. Assign u and dv: ???? = ???? , ???????? = ???? ????????, ???????? = sin ???? ????????, ???? = −cos(????) ???? ( ) ???? 2. Substitute into equation: −???? cos ???? + ∫ cos(????)???????? 3. Must do integration by parts again so assign u and dv: ???? = ???? , ???????? = ???? ????????, ???????? = cos ???? ????????, ???? = sin(????) ???? ???? 4. Substitute into equation again: −???? cos ???? + ???? sin ???? − ∫ ???? sin(????)???????? 5. Since we end up with the same integral again, just add it to original side of the equation and divide by tw∫:???? sin(????)???????? = ???? ???? ???? −???? cos ???? + ???? sin ???? − ???? si∫(????)???????? becomes 2 ∫ sin(????)???????? = −???? cos ???? + ???? sin ???? and then becomes the −???? cos ???? +???? sin ???? solution:∫ ???? sin(????)???????? = 2 e. Fundamental Theorem of Calculus: ∫ ???? ???? ???????? = ????(????) f. Calculus with Position (x), Velocity (v), and Acceleration (a) Functions ???? ???? = ∫ ???? ???? ???????? = ∬ ???? ???? ???????? ???????? ???? ???? ???? ???? = ???????? = ????????2 2. Ordinary Differential Equations a. Introduction: Solution to an algebra problem is a value or set of values Solution to a differential equation is a function or set of functions General solution: form some solutions to a differential equation take Specific solution: solution that adheres to give initial values Differential Equation: any equation with a differential in it b. Directional Fields (Slope fields) Graph of slopes at different points Look at possible solutions and end behavior c. Separable Differential Equations: Any differential equation that can be separated by variable by only multiplying or dividing To solve: 1. Separate variables 2. Integrate 3. Solve for y Will not always be possible May have to worry about interval of validity Most solutions will not be valid for all x 3 Example: ???? =′ ???????? √1+????2 ???????? ????????3 1. ???????? = √ 1+????2 −1 2. ???? −3???????? = ???? 1 + ???? 2) 2???????? −1 3. ∫ ????−3 ???????? = ∫???? 1 + ???? 2 ) 2???????? 1 2 4. − 2????2 = √ + ???? + ???? 1 5. ????(????) = ±√− 2 1+???? +???? d. Linear Differential Equations: ???????? Take the form: ????????+ ???? ???? ???? = ????(????) To solve: 1. Use integrating factor ???? ???? = ???? ∫???? ???? ???????? 2. Multiply both sides of equation by ???? ???? and simplify (takes form of inverse product rule: ???? ???? ????????+ ???? (????)???? = (???? ???? ???? ???? )′) ???????? 3. Integrate both sides and remember constants of integration 4. Divide or multiply to get desired function ???????? Example: ???????? = 9.8 − 0.196???? ???????? ????????+ 0.196???? = 9.8 ∫0.196 ???????? 0.196???? ???? ???? = ???? = ???? ????0.196????????????+ ???? 0.196(0.196????) = 9.8???? 0.196???? ???????? ???? 0.196???? )′= 9.8???? 0.196???? 0.196???? ′ 0.196???? ∫(???? ???? ???????? = ∫ 9.8???? ???????? 0.196???? 0.196???? ???? ???? = 50???? + ???? ???? ???? = 50 + ???????? 0.196???? e. Models of Motion No Air Resistance: 1. ???? ???? = −???? 2. ???? ???? = −???????? + ???? 1 ????????2 3. ???? ???? = − + ???? 1 + ???? 2 2 4. ???? ???????????????????????????? occurs at ???? = 0 5. (???? ???? can also be written as: ???? ???? = −???????? + ???? if given ???? 0 = ????) 2 0 0 and ????(????) can be rewritten as ???? ???? = − ???????? + ???? ???? + ???? if given 2 0 0 ???? 0 = ???? ) 0 Air Resistance Proportional to Velocity: 1. Terminal velocity: − ???????? ???? 2. Air Resistance Proportional to the Square of Velocity: 2 1. When v is positive ???? ???? = −???????? : a. 2 2. When v is negative ???? ???? = ???????? : a. f. Exact Differential Equations: ???????? ???????? Differential of a function ????(????,????) is ???????????????????? + ???????????????? Expression of form: ???? ????,???? ???????? + ???? ????,???? ???????? is exact if it’s the differential ???????? ???????? of ????(????,????) where ???????? = ????(????,????) and ???????? = ????(????,????) ???????? ???????? In other words, it’s exact =f ???????? ???????? ???? ???? ???????? ???????? ???? ???? ???????????????? = ???????? = ???????? = ???????????????? To solve: 1. Check if exact with ???????? = ???????? ???????? ???????? a. If exact: i. Integrate ???? ????,???? ???????? and add ????(????) (OR Integrate ???? ????,???? ???????? and add ????(????)) to get ???? ????,???? = ) ∫ ???? ????,???? ???????? + ????(????) (OR ???? ????,???? = ) ∫ ???? ????,???? ???????? + ????(????)) ii. Solve for ????(????) by setting ???? ????,???? = ????(∫???? ????,???? ????????+????(????)) ( ) ????(∫???? ????,???? ????????+????(????)) ???????? (OR ???? ????,???? = ???????? ) iii. Example: 2???? − 1 ???????? + 3???? + 7 ???????? = 0 1. ???????? = 0 = ???????? = 0 ???????? ???????? 2. ∫ (2???? − 1 ???????? + ???? ???? = ???? + ???? + ???? ???? ( ) 2 ????(???? +????+???? ???? ) 3. ???????? = 3???? + 7 ( ) 4. 3???? + 7 = 0 + 0 + ????′ ???? 5. ∫ 3???? + 7 ???????? = ∫????′ ???? ???????? ′ ????) 3 2 6. ???? = ????2+ 7???? 7. ???? ????,???? =) ∫ ???? ????,???? ???????? + ???? ???? so ???? ????,???? = ) ???? + ???? + ???? + 7???? = ???? 2 8. Solution: ???? + ???? + ???? + 7???? = ???? 2 b. If not exact and given an integrating factor: i. Multiply integrating factor by both sides, then solve c. If not exact and not given an integrating factor: i. Find integrating factor: 1. If ℎ = ( ???????? − ????????) depends only on x, then ???? ???????? ???????? use ???? ???? = ???? ∫ ℎ ???? ???????? 1 ???????? ???????? 2. If ???? = ( − ) depends only on x, then ???? ???????? ???????? use ???? ???? = ???? − ∫ ???? ???????? ii. Multiply integrating factor by both sides, then solve g. Existence and Uniqueness of Solutions: ′ IVP ???? = ????(????,????) and ???? ????( 0 = ????0has a solution if ????(????,????) is continuous on rectangle R on tx-plane and (0 ,0 ) is a point in R 1. ???? = {???? ≤ ???? ≤ ????, ???? ≤ ???? ≤ ????} 2. Will be defined until curve t leaves R (but CAN still be defined after that) Unique if: 1. ????(????,????) is continuous in R ???????? 2. is continuous in R ???????? 3. (???? 0???? 0 is in R 4. There are solutions ???? ???? and ????(????) that satisfy ???? ???? ) = ???? = ???? ????( ) 0 0 0 If two solutions go through the same point where ???????? is defined, they are ???????? the same function ′ 2 4 ( ) Example: ????(????) is a solution to ???? = ???? − ???? + 2???? with ???? 1 = 0. Prove ???? ???? < ???? for all t where x is defined. 1. Function is defined and continuous everywhere: solution exists 2. ???????? = 2???? which is continuous everywhere: solution is unique ???????? 3. Look at ???? 1 = ???? and check that it’s a solution (it is) 4. Plug in 1 for t to get ???? 1 = 1 or (1,1) 1 5. Since we want (1,0), and (1,0) is under the parabolic solution that contains (1,1), (???? 1 < ???? (1)), then ???? ???? < ???? (????) so ???? ???? < ???? 2 1 1 6. Since it’s unique, they don’t cross so ????(????) must always be below ???? ???? = ???? (for all t where x is defined) 1 h. Autonomous and Equilibrium Equations: Autonomous equations: equations only involving original variable (???? = ′ ????(????)) ???? = ????(????), ???? ????( )0 = 0, if ???? ???? = ???? ,0it’s an equilibrium solution ′ 3 2 Example: ???? = −???? − ???? + 2???? + 2 1. Set equal to zero and find zeros: ???? = −1,± 2 √ 2. Draw Phaseline (circle: unstable, dot, stable) 3. 4. Another stability test: 1. Suppose ???? is equilibrium point for ???? = ???? ???? . Then if: ′ 0 a. ???? ???? > 0, ???? is u0stable b. ???? ???? < 0, ???? is a0ymptotically stable ′ c. ???? ???? = 0, no conclusion i. Population Growth Malthusian Model: reproductive rate proportional to population 1. Logistic Model: r = reproductive rate, k = carrying capacity 1. 3. Matrix Algebra a. b.

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