Ordinary Differential Equations Everything is ordinary about them Popping tags means A. Popping bubble wrap B. Using firecrackers C. Changing tags of regular items in a store with tags from clearance items D. Taking illicit drugs Popping bubble wrap Using firecrackers Changing tags of regular... Taking illicit drugs How long will it take to cool the trunnion? Physical Examples d mc ha( ), a (0) room
Ordinary Differential Equations Problem: The trunnion initially at room temperature is put in a bath of dry-ice/alcohol. How long do I need to keep it in the bath to get maximum contraction ( within reason )? Assumptions The trunnion is a lumped mass system. a. What does a lumped system mean? It implies that the internal conduction in the trunnion is large enough that the temperature throughout the ball is uniform. b. This allows us to make the assumption that the temperature is only a function of time and not of the location in the trunnion. Energy Conservation Heat Lost Heat In Heat Lost = Heat Stored Rate of heat lost due to convection= ha(t-t a ) h = convection coefficient (W/(m.K)) A = surface area, m T= temp of trunnion at a given time, K
Heat Stored Heat stored by mass = mct where m = mass of ball, kg C = specific heat of the ball, J/(kg-K) Energy Conservation Rate at which heat is gained Rate at which heat is lost =Rate at which heat is stored 0- ha(t-t a ) = d/(mct) 0- ha(t-t a ) = m C dt/ Putting in The Numbers Length of cylinder = 0.65 m Radius of cylinder = 0.3 m Density of cylinder material = 7800 kg/m 3 Specific heat, C = 450 J/(kg-C) Convection coefficient, h= 90 W/(m -C) Initial temperature of the trunnion, T(0)= 7 o C Temperature of dry-ice/alcohol, T a =-78 o C The Differential Equation Surface area of the trunnion A=rL+r =**0.3*0.65+**0.3 =.744 m Mass of the trunnion M= V = (r L) = (7800)*[*(0.3) *0.65] = 378 kg 3
The Differential Equation ha ( T Ta ) mc dt 90.744 ( T 78) 378 450 dt.530 T (0) 7 4 ( T 78)), dt Temperature in Celcius 40 0 0-0 -40-60 Solution Exact and Approximate Solution of the ODE by Euler's Method Exact Approximation -80 0 000 000 3000 4000 5000 6000 7000 8000 9000 0000 Time in seconds Time Temp (s) ( o C) 0 7 000 0.4 000-9.4 3000-34.5 4000-45.3 5000-53.59 6000-59.77 7000-64.38 8000-67.83 9000-70.40 0000-7.3 END What did I learn in the ODE class? 4
In the differential equation x 3y e, y(0) 6 the variable x is the variable In the differential equation x 3y e, y(0) 6 the variable y is the variable A. Independent B. Dependent A. Independent B. Dependent A. B. A. B. Ordinary differential equations can have these many dependent variables. Ordinary differential equations can have these many independent variables. A. one B. two C. any positive integer 33% 33% 33% A. one B. two C. any positive integer 33% 33% 33% A. B. C. A. B. C. 5
A differential equation is considered to be ordinary if it has A. one dependent variable B. more than one dependent variable C. one independent variable D. more than one independent variable Classify the differential equation x 4 y e 3, y(0) A. linear B. nonlinear C. undeterminable to be linear or nonlinear 33% 33% 33% linear nonlinear undeterminable... Classify the differential equation x xy e 3, y(0) A. linear B. nonlinear C. linear with fixed constants D. undeterminable to be linear or nonlinear Classify the differential equation y e 3, y(0) A. linear B. nonlinear C. linear with fixed constants D. undeterminable to be linear or nonlinear x 6
The velocity of a bo is given by A. B. C. D. v( t) e e t e t t, t 0 Then the distance covered by the bo from t=0 to t=0 can be calculated by solving the differential equation for x(0) for, x(0) 0, x(0) t e, x(0) 0 t e t, x(0) 0 The form of the exact solution to A. B. C. D. Ae Ae Ae Ae.5x.5 x.5x.5x Be Bxe Be x x x Bxe x 3y e x, y(0) is 3 4 END 8.03 Euler s Method 7
Euler s method of solving ordinary differential equations f ( x, y), y(0) 0 states A. yi yi f ( x, y) h B. yi yi f ( xi, yi ) h C. yi yi f ( xi, yi ) D. y f ( x, y h i i i ) To solve the ordinary differential equation 3 y sin x, y(0), by Euler s method, you need to rewrite the equation as A. B. C. D. sin x y, y(0) (sin x y ), y(0) 3 3 5y ( cos x ), y(0) 3 3 sin 3 x, y(0) The order of accuracy for a single step in Euler s method is A. O(h) B. O(h ) C. O(h 3 ) D. O(h 4 ) The order of accuracy from initial point to final point while using more than one step in Euler s method is A. O(h) B. O(h ) C. O(h 3 ) D. O(h 4 ) O(h) O(h) O(h3) O(h4) O(h) O(h) O(h3) O(h4) 8
END RUNGE-KUTTA 4 TH ORDER METHOD Do you know how Runge- Kutta 4 th Order Method works? A. Yes B. No C. Maybe D. I take the 5 th Yes No Runge-Kutta 4 th Order Method y f ( x, y), y(0) y0 yi k k k 6 k i 3 4 k f x i, y i k f xi h, yi kh k3 f xi h, yi kh k4 f xi h, yi k3h h 9
END FINITE DIFFERENCE METHODS d y d y Given 6x 0.5x, y(0) 0, y() 0, The value of at y(4) using finite difference method and a step size of h=4 can be approximated by A. B. C. D. y( 8) y(0) 8 y( 8) y(4) y(0) 6 y( ) y(8) y(4) 6 y( 4) y(0) 4 END 0