(competition between 2 species) (Laplace s equation, potential theory,electricity) dx4 (chemical reaction rates) (aerodynamics, stress analysis)

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1 SSE1793: Tutorial 1 1 UNIVERSITI TEKNOLOGI MALAYSIA SSE1793 DIFFERENTIAL EQUATIONS TUTORIAL 1 1. Classif each of the following equations as an ordinar differential equation (ODE) or a partial differential equation(pde), give the order, and indicate the independent and dependent variables. If the equation is an ODE, indicate whether the equation is linear or nonlinear. (a) 3 d x x = cos 3t (b) ( 3x) = x(1 3) (c) u x + u = 0 (mechanical vibration, electrical circuit, seismolog) (competition between species) (Laplace s equation, potential theor,electricit) (d) dp = kp(p p) where P and p are constants (logistic curve, epidemiolog,economics) (e) = (4 x)(1 x) (f) x d + + x = 0 (g) 8 d4 = x(1 x) 4 (chemical reaction rates) (aeronamics, stress analsis) (deflection of beams). Determine whether the given equation is separable, linear, neither or both. = sin x +. b. x + t x = sin t. e. (t + 1) = t. = ex+ x +. d. 3t = e t + lnt. 3. Solve the following ODE using separable variables. = sec b. x dv 1 + x = 1 4v 3v + x = x d. = 3x (1 + ) e. 1 + e cos x sin x = 0 f. (x + x ) + e x = 0 4. Solve the following Initial Value Problem. = x 3 (1 ), (0) = 3 b. = + 1(cosx), (π) = 0 d. = (1 + )tanx, (0) = 3 = x cos, (0) = π 4 5. Obtain the general solution to the following ODE. dr + r tan θ = sec θ b. (t + + 1) = 0 dθ (x + 1) + x = x d. (x + 1) = x + x 1 4x 6. Solve the following Initial Value Problem. + 4 e x = 0, (1) = e 1 b. + 3 x + 4 e x = 0, (0) = = 3x, (1) = 1 d. sin x + cos x = x sinx, (π ) =

2 7. Classif the equation as separable, linear, exact or none of these. Notice that some equations ma have more than one classifications. (x + x 4 cosx) x 3 = 0. b. (x 10 3 ) + x = 0. + (3 + x x ) = 0. d. + (x + cos ) = 0. e. θdr + (3r θ 1)dθ = 0 8. Classif the equation as separable, linear, exact or none of these. Notice that some equations ma have more than one classifications. (a) (x + 3) + (x 1) = 0. (b) (cos x cos + x) (sin x sin + ) = 0. (c) t + (1 + ln) = 0. (d) e t ( t) + (1 + e t ) = 0. (e) (x x ) + ( x 1 + x ) = 0 9. Solve the initial value problem: (a) (e x 1 ) + (xex + x ) = 0, (1) = 1. (b) ( sin x) + ( 1 x ) = 0, (π) = 1. x 10. For each of the following equations, find the most general function M(x, ) or N(x, ) respectivel so that the equation is exact. (a) M(x, ) + (sec x ) = 0. (b) ( cos (x) + e x ) + N(x, ) = Consider the equation ( + x) x = 0 (a) Show that this equation is not exact. (b) Show that multipling both sides of the equation b ields anew equation that is exact. (c) Use the solution of the resulting exact equation to solve the original equation. (d) Were an solutions lost in the process? 1. Use the method discussed under Homogeneous Equations to solve: (a) (3x ) + (x x 3 1 ) = 0. (b) (x + ) + x = 0.. (c) dθ = θ sec ( θ ) +. θ (d) (ln lnx + 1) =. x 13. Use the method discussed under Equations of the form = G(ax + b) to solve: = x + 1. b. = (x + 5).

3 14. Use the method discussed under Bernoulli Equations to solve: = ex 3. b. + x = 5(x ) 1. = x x. d. + tx3 + x t = Newton s Law of Cooling. According to Newton s Law of Cooling, if an object at temperature T is immersed in a medium having the constant temperature M, then the rate of change of T is proportional to the difference of temperature M T. This gives the differential equation, dt = k(m T) (a) Solve the equation for T. (b) A thermometer reading 100 o is placed in a medium having the constant temperature of 70 o. After 6 minutes, the thermometer reads 80 o. What is the reading after 0 min? (c) Blood plasma is stored at 40 o. Before the plasma can be used, it must be at 90 o. When the plasma is placed in an oven at 10 o, it takes 45 min for the plasma to warm to 90 o. How long will it take for the plasma to warm to 90 o if the oven is set at 100 o, 140 o and 80 o respectivel? (d) It was noon on a cold December da in Cameron Highland; 16 o C. Detective Musa arrive at the crime scene to find the sergeant leaning over a bo. The sergeant said that there were several suspects. If onl the knew the exact time of dealth, then the could narrow down the list. Detective Musa took out a thermometer and measured the temperature of the bo; 34.5 o C. He then left for lunch. Upon returning at 1:00 pm, he found the bo temperature to be 33.7 o C. When did the murder occur? Hint: Normal bo temperature is 37 o C. (e) Just before midda, the bo of an apparent homicide victim is found in a room that is kept at a constant temperature of 70 o F. At 1 noon, the temperature of the bo is 80 o F and at 1 pm it is 75 o F. Assume that the temperature of the bo at the time of death is was 98.6 o F and that it has cooled in accord with Newton s law of cooling. What was the time of death? 16. Free Fall. An object falls through the air toward earth. Assuming onl air resistance and gravit are acting on the object, it is found that the velocit v must satisf the equation m dv = mg bv where m is the mass, g is the acceleration due to gravit, and b > 0 is a constant. If m = 100 kg, g = 9.8 m/sec, b = 5 kg/sec, and v(0) = 10m/sec, solve for v(t). What is the limiting (i.e., terminal) velocit of the object. 17. Vertical Motion. A particle moves verticall under the force of gravit against air resistance kv, where k is a constant. The velocit v at an time t is given b the differential equation dv = g kv. If the particle starts off from rest show that such that λ = v = λ(eλkt 1) (e λkt + 1) g. Then find the velocit as the time approaches infinit. k 3

4 18. Electric Circuit. The simplest electric circuit shown in Figure 1 contains an electromotive force (usuall a batter or generator) that produces a voltage of E(t) volts (V) an a current of I(t) amperes (A) at time t. The circuit also contains a resistor with a resistance of R ohm Ω and an inductor with an inductance of L henries (H). Ohm s Law gives the drop in voltage due to the resistor as RI. The voltage drop due to the inductor is L di. One of Kirchhoff s sas that the sum of voltage drops is equal to the supplied voltage E(t). Thus we have L di + RI = E(t) which is a first order linear differential equation. The solution gives the current I at time t. (a) Suppose that in the simple circuit of Figure 1, the resistance is 1Ω and the inductance is 4H. If a batter gives a constant voltage of 60V and the switch is closed when t = 0, so the current starts with I(0) = 0, find i. I(t) ii. the current after 1 sec iii. the limiting value of the current. (b) Suppose that the resistance and inductance remain as in part (a) but, instead of the batter, we use a generator that produces a variable voltage of E(t) = 60 sin 30t volts. Find I(t). 19. Electric Circuit. Figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads (F), and a resistor with a resistance of R ohms (Ω). The voltage drop across the capacitor is Q/C where Q is the charge (in coulombs), so in this case Kirchhoff s Law gives but I = dq, so we have R RI + Q C = E(t) dq + 1 C Q = E(t). (a) Suppose the resistance is 5Ω, the capacitance is 0.05F, a batter gives a constant voltage of 60V, and the initial charge is Q(0) = 0C. Find the charge and the current at time 4t. (b) In the circuit of part (a), R = Ω, C = 0.01F, Q(0) = 0 and E(t) = 10 sin60t. Find the charge and current at time t. 4

5 UNIVERSITI TEKNOLOGI MALAYSIA SSE1793 DIFFERENTIAL EQUATIONS ANSWERS TO TUTORIAL 1 1. (a) ODE, nd order, ind.var. t, dep.var. x, linear. (b) ODE, 1 st order, ind.var.x, dep.var., nonlinear. (d) ODE, 1 st order, ind.var.t, dep.var.p, nonlinear.. linear b.separable not linear, not separable d.linear e.separable and linear 3. (a) + sin = 4 arctanx + C. (c) x = Cet Ce t 1, x = 1. 1 (e) = C e cos x. 4. (a) = e x (c) = sin x + sinx. (d) = arctan (1 + x ). r = sinθ + C cos θ. b. = t + Ce t. = 1 + C ( x + 1 ) (b) x = t 1 t linear with as dependent variable d. exact, linear with x as dep. var e. linear, r as dep var. (C 3x) = (x 1). b. sin x cos + x = C. t ln + t = C. e. x + arctan (x) = C. 9(b). 10(a). sin x x cos x = ln π 1. (equation is separable, not exact.) ln + f(x) 11. = x C x. ( ) 1(a). ln x 6 x = C d. es. (x + C) 13. (a) = x and = x. 4 (b) = x + (6 + 4Cex ) and = x + 4. (1 + Ce x ) 14. (b) = 5x and = 0. x 5 + C (d) x = t ln t + Ct and x = 0. 15(c). 8. min; 31.8 min; Never attains desired temperature