Methods of Solving Linear Differential Equations and Applications

1. ENGG2013 Unit 24Linear DE and Applications Apr, 2011.

2. Outline • Method of separating variable • Method of integrating factor • System of linear and first-order differential equations • Graphical method using phase plane kshum

3. Nomenclatures • “First-order”: only the first derivative is involved. • “Autonomous”: the independent variable does not appear in the DE • “Linear”: • “Homogeneous” • “Non-homogeneous” c(t) not identically zero kshum

4. Separable DE • “Separable”: A first-order DE is called separable if it can be written in the following form • Examples • x’ = cos(t) • x’ = x+1 • x’ = t2sin(x) • t x’ = x2–1 • All linear homogeneous DE kshum

5. SEPARABLE DE ANDMETHOD OF SEPARATING VARIABLES kshum

6. How to solve separable DE • Write x’= f(x) g(t) as . • Separate variable x and t (move all “x” to the LHS and all “t” to the RHS) • Integrate both sides kshum

7. Example Solve (1) Write the DE as (2) Separate the variables (3) Integrate both sides General solution to x’=t/x kshum

8. Solution curves • The solutions are hyperbolae Sample solutions Some constant kshum

9. Example: Newton’s law of cooling • Suppose that the room temperature is Tr = 24 degree Celsius. The temperature of a can of coffee is 15 oC at T=0 and rises to 16 oC after one minute. • T(0) = 15, T(1) = 16. • Find the temperature after 10 minutes Proportionality constant kshum

10. LINEAR NON-HOMOGENEOUS DE METHOD OF INTEGRATING FACTOR kshum

11. Example: RC in series • Physical laws • Voltage drop across resistor = VR(t) = R I(t) • Voltage drop across inductor = C VC(t) = Q(t) Charge From Kirchoff voltage lawVC(t) + VR(t) = sin(t) Linear non-homogeneous kshum

12. Linear DE in standard form • Linear equation has the following form • By dividing both sides by p(t), we can write the differential equation in standard form kshum

13. Product rule of differentiation • Idea: Given a DE in standard form Multiply both sides by some function u(t) so that the product rule can be applied. kshum

14. Illustrations • Solve the initial value problem • Find the general solution to kshum

15. Example: Mixing problem • In-flow of water: 10 L per minute • Out-flow of water: 10 L per minute • In-flowing water contains Caesium with concentration 5 Bq/L • Describe the concentration of Ce in the water tank as a function of time. Water tank 1000 L Initial Caesium concentration = 1 Bq/L kshum

16. Henri Becquerel http://en.wikipedia.org/wiki/Henri_Becquerel • French physicist • Dec 1852 ~ Aug 1908 • Nobel prize laureate of Physics in 1903 (together with Marie Curie and Pierre Curie) for the discovery of radioactivity. • Bq is the SI unit for radioactivity • Defined as the number of nucleus decays per second. kshum

17. Back to the RC example • Write it in standard form • Multiply by an unknown function u(t) kshum

18. Integrating factor • Is there any function u(t) such that u’(t) = u(t)/RC ? •  Choose u(t) = exp(t/RC) kshum

19. Now we can integrate Use a standard fact from calculus kshum

20. Solution to RC in series • General solution • If it is known that Q(0) = 0, then approaches zero as t   Steady-state solution kshum

21. Sample solution curves • Take R=C = 1, =10 for example. Different solutions correspond to different initial values. Steady state Transient state kshum

22. SYSTEM OF DIFFERENTIAL EQUATIONS kshum

23. Interaction between components • If we have two or more objects, each and they interact with each other, we need a system of differential equations. • Metronomes synchronization • http://www.youtube.com/watch?v=yysnkY4WHyM • Double pendulum • http://www.youtube.com/watch?v=pYPRnxS6uAw kshum

24. General form of a system of linear differential equation • System variables: x1(t), x2(t), …, xn(t). • A system of DE Some functions kshum

25. System of linear constant-coeff. differential equations • System variables: x1(t), x2(t), x3(t). • Constant-coefficient linear DE • aij are constants, • g1(t), g2(t) and g3(t) are some function of t. • Matrix form: kshum

26. Application 1: Mixing f12 • C1(t) and C2(t) are concentrations of a substance, e.g. salt, in tank 1 and 2. • Given • Initial concentrations C1(0) = a, C1(0) = b. • In-low to tank 1 = f1m3/s, with concentration c. • Flow from tank 1 to tank 2 = f12m3/s • Flow from tank 2 to tank 1 = f21m3/s • Out-flow from tank 2 = f2m3/s • Objective: Find C1(t) and C2(t). Water tank 2 Volume = V2 m3 Concentration = C2(t) Water tank 1 Volume = V1 m3 Concentration = C1(t) f2 f1 f21 kshum

27. Modeling • Consider a short time interval [t, t+t] • C1 = C1(t+t)–C1(t) = cf1t + f21C2t – f12C1t • C2 = C2(t+t)–C2(t) = f12C1t– f21C2t – f2C2t • Take t  0, we have C1’ = – f12C1+ f21C2+ cf1 C2’ = f12C1 – (f21+ f2) C2 kshum

28. Graphical method • For autonomous system, • we can plot the phase plane (aka phase portrait) to understand the system qualitatively. • Select a grid of points, and draw an arrow for each point. The direction of each arrow is kshum

29. Phase Plane C1’ = – 6C1+ C2+ 10 C2’ = 6C1 – 6 C2 • Suppose • f1 = 5 • f2 = 5 • f12 = 6 • f21 = 1 • c = 2 • Initial concentrationsare zero Converges to (2,2) kshum

30. Convergence C1’ = – 6C1+ C2+ 10 C2’ = 6C1 – 6 C2 • (C1,C2)=(2,2) is a critical point. • C1’ and C2’ are both zero when C1= C2=2. • The analyze the stability of critical point, we usually make a change of coordinates and move the critical point to the origin. • Let x1 = C1–2, x2 = C2–2. x1’ = – 6x1+ x2 x2’ = 6x1 – 6 x2 kshum

31. Phase plane of a system with stable node All arrows points towardsthe origin kshum

32. Sample solution curves The origin is a stable node kshum

33. Theoretical explanation for convergence • The eigenvalues of the coefficient matrix are negative. Indeed, they are equal to –3.5505 and –8.4495. • The corresponding eigenvectors are [0.3780 0.9258] and [–0.3780 0.9258] kshum

34. Eigen-direction • If we start on any point in the direction of the eigenvectors,the system converges to the critical point in a straight line. • This is another geometric interpretation of the eigenvectors. kshum

35. Application 2: RLC mesh circuit • Suppose that the initial charge at the capacity is Q0. • Describe the currents in the two loops after the switch is closed. i1(t) i2(t) • Physical Laws • Resistor: V=R i • Inductor: V=L i’ • Capacitor: V=Q/C • KVL, KCL Homework exercise kshum

36. An expanding system • Both eigenvalues are positive. kshum

37. Phase Plane of a system with unstable node The origin is an unstable node. The red arrows indicate the eigenvectors kshum

38. A system with saddle point • One eigenvalue is positive, and another eigenvalue is negative kshum

39. Phase Plane of a system with saddle node The origin is a saddle point. The thick red arrows indicatethe eigenvectors kshum

40. Conclusion The convergence and stability of a system of linear equations is intimately related to the signs of eigenvalues. kshum