dx. Ans: y = tan x + x2 + 5x + C

Transcription

1 Chapter 7 Differential Equations and Mathematical Modeling If you know one value of a function, and the rate of change (derivative) of the function, then yu can figure out many things about the function. One of the early accomplishments of calculus was predicting the future of a planet from its present position and velocity. 7. Slope Fields and Euler s Method- Def: An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation. Find all functions y that satisfy dy sec 5. Ans: y = tan C General Solution Initial Value Problem: To Solve an Initial Value Problem:. Cross-Multiply and separate variables (keep y on the left side). Integrate 3. Substitute in your initial condition 4. Simplify. Final answer is y = Find the particular solution to the equation dy ( e 6 ) dy ( e 6 ) y = e 3 + C. dy e 6 whose graph passes through (, 0) Apply the initial position: 0 = e () 3 + C C = e Answer: y = e 3 + e.

2 Find the particular solution to the equation dy dy y dy y (3 ) y (3 ) (3 ) 3 ln y c e c 3 y y Ce 3 Sub in Initial Condition: Ce 03, so C =. Answer: y e 3 dy 3y with y(0)= Slope Fields- A field of little line segments, each of which represents the slope of the tangent to the solution of a differential equation. (You are drawing little tangent lines.) Sketch a slope field for: dy (-, ) - (-, ) - (0, ) 0 (, ) (-, ) - (-, ) - (0, ) 0 (, ) (-, 0) - (-, 0) - (0, 0) 0 (, 0) (-, -) - (-, -) - (0, -) 0 (0, -) (-, -) - (-, -) - (0, -) 0 (0, -)

3 Solve the differential equation above: dy dy y C dy Euler s Approimation Method- We use a series of tangent lines to find numerical approimations to solutions of differential equations. Recall Linear Approimation: L()=f(a) + f`(a) ( a) dy Given the differential equation step size Δ =. to approimate y at =.3, with initial point (0, ). Use Euler s method, with y = y 0 + dy y = y + dy y 3 = y + dy 0 0 (Δ), so y = + (0) (.) y, (0,) y, (.,) (Δ), so y = + (.) (.) (Δ), so y 3 =.0 + (.) (.) 3 3 Euler s Approimation is.06 y, (.,.0) y, (.3,.06)

4 7. Antidifferentiation by Substitution- Properties of Indefinite Integrals kf ( ) k f( ) ( f( ) g( )) f( ) g( ) Power Formulas ( ) n u n n u du u C n u du du lnu C Trigonometric Formulas cos udu sin uc sin udu cos uc sec udu tan uc csc udu cot uc sec utan udu sec uc csc ucot udu csc uc Eponential and Logarithmic Formulas u u u u e du e C a a du C lna lnudu uln uuc log lnu ln audu du udu ulnu u ln ln ln C a a a

5 U-Substitution In Indefinite Integrals: 5 u= + du = u du u C C tan = sin cos u = cos du lnuc u lncos C du = - sin - du = sin 4 u = 4 + du = 4 ¼ du = 3 u du u du u C C = 3 U-Substitution in Definite Integrals: 3 tan sec u = tan 0 du = sec Change your bounds. When = 0, u(0) = 0; when 3 udu u C = , u tan 3 3 3

6 7 u = + 0 du = Changing Bounds: u (0) = and u (7) = 9 9 du u 9 9 ln u ln9 ln or ln 7.3 Antidifferentiation by Parts- udv uv vdu Let u be something you can simplify. Choices for u (LIPET) L: Natural Log I: Inverse Trig P: Polynomial E: Eponential T: Trig cos u = du = cos sin sin sin cos C dv = cos v = sin

7 ln ln ln ln ln C u = ln du dv = v = 7.4 Eponential Growth and Decay- If we are interested in a quantity y (population, money, radioactive elements, etc) that increases or decreases at a rate proportional to the amount present dy ky dt and y = y 0 when t = 0 Differential Equation Initial Condition dy dt If y is positive and increasing 0, then k > 0. Rate of Growth is proportional to what has already been accumulated. dy dt If y is positive and decreasing 0, then k < 0. Rate of Decay is proportional to the amount still left.

8 dy ky dt dy kydt dy kdt y ln ykt c kt c e y kt yce Law of Eponential Change: If y changes at a rate proportional to the amount present and y = y 0 at time t = 0, then: y ye 0 kt k > 0 growth k < 0 dy ky dt, decay k is the rate constant 7.5 Logistic Growth- Populations cannot sustain eponential growth for a very long time. Logistic growth, which starts off eponentially and then changes concavity to approach a maimal sustainable population, is a better model for real-world populations. dp kp Eponential Growth dt dp kpm ( P ) Logistic Differential Equation dt M = Maimum Capacity Population is growing the fastest when it is half the Maimum Capacity

9 Eample: The growth rate of a population P of bears in a newly established wildlife preserve is modeled by the differential equation dp P(00 P) dt, where t is measured in years. a) What is the carrying capacity for bears in this wildlife preserve: 00 bears (Maimum carrying capacity) b) What is the bear population when the population is growing the fastest? The bear population is growing the fastest when it is half the carrying capacity 50 bears Partial Fractions 4 65 cannot be integrated by any of the techniques learned so far. We need to rewrite the fraction another way so that it can be integrated. A B Multiply by LCD 5 A ( ) B ( 5) 4 Solving for A and B, A = and B = ln 5 ln or ln 5 C

10 3 A B Multiply by LCD (+)(-) A (+) + B ( ) =. A = ½ and B = - ½ / / d ln ln ln 3 3 = ln ln ln /4 ln 4 3 / 3

11