Millersville University Department of Mathematics MATH 365 Ordinary Differential Equations January 23, 212 The most up-to-date version of this collection of homework exercises can always be found at http://banach.millersville.edu/ bob/math365/mmm.pdf. 1. Find the general solution to the following ordinary differential equation. dx = (y 1)(x 2)(y + 3) (x 1)(y 2)(x + 3) 2. Find the general solution to the following ordinary differential equation. x 3 e 2x2 +3y 2 dx y 3 e x2 2y 2 = 3. Find the general solution to the following ordinary differential equation. du ds = U + 1 s + su 4. Find the solution to the following initial value problem. dr dφ = sin φ + e2r sin φ 3e r + e r cos 2φ ( π r = 2) 5. A particle moves along the x-axis so that its velocity is proportional to the product of its instantaneous position x (measured from x = ) and the time t (measured from t = ). If the particle is located at x = 54 when t = and x = 36 when t = 1, where will it be when t = 2? 6. Show that the differential equation dx = 4y2 x 4 4xy is non-separable but becomes separable on changing the dependent variable from y to v = y/x. Use this change of variable to find the solution to the original equation. 7. Solve (2y 2 + 4x 2 y) dx + (4xy + 3x 3 ) = given that there exists an integrating factor of the form x p y q where p and q are constants.
8. Solve the equation = αy βyn dx where α, β, and n are constants and n 1. 9. Show that the differential equation can be solved by letting v = ln y. + P(x)y = Q(x)y ln y dx 1. Solve the differential equation [ ] [ ] y (x + y) 1 x dx + 1 =. 2 (x + y) 2 11. Determine the most general function N(x, y) such that is exact, and obtain its solution. (y sin x + x 2 y x sec y) dx + N(x, y) = 12. Solve the differential equation ( 2x sin y x + 2x tan y x y cos y x y y ) ( sec2 dx + x cos y x x + x y ) sec2 =. x 13. Show that the ordinary differential equation yf(xy) dx + xg(xy) = is not exact in general but becomes exact on multiplying by the integrating factor 1 xy(f(xy) g(xy)). 14. Solve the differential equation 15. Solve the differential equation 16. Solve the differential equation dx = 3y2 cot x + sin x cosx. 2y (2x + 2xy 2 ) dx + (x 2 y + 2y + 3y 3 ) =. (2y 2 + 4x 2 y) dx + (4xy + 3x 3 ) = given that there exists an integrating factor of the form x p y q, where p and q are constants.
17. Show that a ball thrown vertically upward with initial velocity v takes twice as long to return as to reach the highest point. Find the velocity upon return. Ignore air resistance. 18. A bo moves in a straight line with constant acceleration a. If v is the initial velocity, v is the instantaneous velocity, and s is the distance traveled after time t, show that (a) v = v + at, (b) s = v t + 1 2 at2, (c) v 2 = v 2 + 2as. 19. A mass m is thrown upward with initial velocity v. Air resistance is proportional to its instantaneous velocity with constant of proportionality k. Show that the maximum height attained is ( mv k m2 g ln 1 + kv ). k 2 mg 2. According to Einstein s special theory of relativity, the mass of a particle varies with its velocity v according to the formula m = m 1 v2 c 2 where m is the rest mass and c is the speed of light. The differential equation of motion is d F = m v. dt 1 v2 c 2 If a particle starts from rest at t = and moves in a straight line acted upon only by a constant force F, what distance will it cover and what will its velocity be at time t? Show that as t, v c. 21. A chemical C is to be dissolved in water. Experimentally, the rate at which C enters the solution varies as the product of (i) the instantaneous amount of C which remains undissolved, (ii) the difference between the instantaneous concentration of the dissolved chemical and the maximum concentration possible at the given conditions of temperature and pressure (this maximum occurs when the solution is saturated and further increase of the chemical dissolved is not possible). If 5 lb of C are placed in 2 gal of water, it is found that 1 lb dissolves in 1 hr. Assuming a saturated solution to have a concentration of 4 lb/gal: (a) how much of chemical C remains undissolved after 4 hrs? (b) what is the concentration of the solution after 3 hrs?
(c) when will the concentration be 2 lb/gal? 22. Uranium disintegrates at a rate proportional to the amount present at any instant. If M 1 and M 2 grams are present at times T 1 and T 2 respectively, show that the half-life is (T 2 T 1 ) ln2 ln(m 1 /M 2 ) 23. In a certain culture, bacteria grow at a rate which is proportional to the pth power of the number present. If the number present is N initially, and if after time T 1 the number is N 1, show that the number is N 2 in a time given by Discuss the cases where p = and p = 1. T 2 = N1 p 2 N 1 p N 1 p 1 N 1 p. 24. A radioactive isotope having a half-life of T minutes is produced in a nuclear reactor at the rate of a grams per minute. Show that the number of grams of the isotope present after a long time is at/ ln 2. 25. Suppose the maximum concentration of a drug present in a given organ of constant volume V must be c max. Assuming that the organ does not contain the drug initially, that the liquid carrying the drug into the organ has constant concentration c > c max, and that the inflow and outflow rates are both equal to b, show that the liquid must not be allowed to enter for a time longer than V b ln ( c c c max 26. Can y = e rx for constant r be used to solve the ordinary differential equation, You must explain your answer. ). y xy + y =? 27. Find the general solution to the following ordinary differential equation. y (4) 2y + 4y = 28. Find the general solution to the following ordinary differential equation. y (4) 2y 16y + 12y + 12y = 29. Find a constant p such that y = x p solves the ordinary differential equation, Find the general solution to the equation. x 2 y + 3xy + y =.
3. Find a constant p such that y = x p solves the ordinary differential equation, (1 x 2 )y 2xy + 2y =. Find the general solution to the equation. 31. Solve the ordinary differential equation y + y = 6 cos 2 x such that y() = and y(π/2) =. 32. Find the general solution of y 3y + 2y = 4 sin 3 (3x). 33. Find the general solution of y + 4y = sin 4 x. 34. Find the general solution of 35. Show that the solution to y + 4y = cosx cos(2x) cos(3x). y + a 2 y = f(t) satisfying initial conditions y() = y () = is y(t) = 1 a t 36. Solve the following ordinary differential equation f(x) sin(a(t x)) dx. y 2y + 4y 8y = 64 sin(2x). 37. A particle moves along the x-axis in such a way that its instantaneous acceleration is given by a = 16e t 2x 8v where x is the instantaneous position measured from the origin, v is the instantaneous velocity, and t is time. If the particle starts from rest at the origin, find its position at any later time. 38. Find the general solution to the ordinary differential equation given that y 1 (x) = cos x is a solution. y + (cos x)y + (1 + sin x)y =
39. Find the general solution to the ordinary differential equation y + (cos x)y + (1 + sin x)y = 1 given that y 1 (x) = cos x is a complementary solution. 4. Show that the equation can be transformed into y + p(x)y + q(x)y = u + f(x)u = by letting y = u(x)v(x) and choosing v(x) appropriately. 41. A weight W suspended from a vertical spring produces a stretch of magnitude a. When the weight is in equilibrium it is acted upon by a force which gives to it a velocity v downward. Show that the weight travels a distance v a/g for a time (π/2) a/g before it starts to return. 42. When a weight at the end of a vertical spring is set into vibration, the period is 1.5 seconds. After adding 8 pounds, the period becomes 2.5 seconds. How much weight was originally on the spring? 43. A spring oscillates vertically. The maximum velocity and acceleration are given, respectively, by v m and a m. Show that the period of oscillation is 2πv m /a m and the amplitude is v 2 m /a m. 44. A spring oscillates with amplitude A and period T. Show that the maximum velocity occurs at the center of the path and has magnitude 2πA/T, while the maximum acceleration occurs at the ends of the path and has magnitude 4π 2 A/T 2. 45. A spring rests taut but unstretched on a horizontal surface. One end is attached to a point O on the surface and the other to a weight W. The weight is displaced so that the spring is stretched a distance a and it is then released. If the coefficient of friction between the weight and the surface is µ, and if the spring constant is k, show that when the spring returns to its unstretched position the magnitude of its velocity is g W (ka2 2µWa) and that this takes time given by [ ( )] W µw π cos 1. gk ka µw
46. If a hole were bored through the earth s center, one would find that an object placed in the hole is acted upon by a force of attraction varying directly as the distance between the object and the earth s center. Assuming the earth is a sphere with a radius of 4 miles, find the time for an object dropped in the hole to return. Find the velocity of the object as it passes through the earth s center. 47. Show that the times at which y(t) = e αt sin(ωt + φ) is a maximum (in magnitude) are given by t n = 1 (tan 1 ω ) ω α + (n 1)π φ for n = 1, 2,.... 48. A mass m is suspended from a vertical spring having spring constant k. At t = the mass is struck so as to give it a velocity v downward. A damping force γv where v is the instantaneous velocity and γ is a positive constant, acts on the mass. The damping is so large that γ > 2 km. (a) Show that the instantaneous position of the mass at time t > is where y(t) = v β e γt/2m sinh βt β = γ2 4km 2m. (b) Show the mass travels downward for time t = 1 β tanh 1 2mβ γ. 49. A mass on a spring undergoes a forced vibration given by m u + k u = A cos 3 ωt. Show that there are two values of ω at which resonance occurs and find them. 5. Consider the initial value problem: Show that solves the initial value problem. m u + k u = F(t) u() = u () =. u(t) = 1 t F(s) sin km k (t s) ds m
51. Solve the following initial value problem using series methods. y + xy 2y = y() = 1 y () = 52. Find the general solution of the following ordinary differential equation using series methods. y 2xy + 4y = 53. Find a solution to the following ordinary differential equation using series methods. y + xy = sin x 54. Solve the following initial value problem using series methods. y + e x y = y() = 1 y () = 55. Find a solution to the following ordinary differential equation using series methods. (cos x)y + (sin x)y = For what values of x does the solution converge? 56. Find the general solution to the following ordinary differential equation. y xy y = 5 x 57. Find the general solution to the following ordinary differential equation. xy + 2y + xy = 2x 58. Find the general solution to the following ordinary differential equation. xy + y + y = 59. Find the general solution to the following ordinary differential equation. xy + xy + y =
6. Consider the ordinary differential equation x 4 y + 2x 3 y + y =. Show that x = is an irregular singular point for this ODE. Let u = 1/x and solve the ODE. 61. Show that J (x) = J 1(x). 62. Show that d dx [xj 1(x)] = xj (x). ( 2 sin x 63. Show that J 3/2 (x) = πx x 2 ( 64. Show that J 3/2 (x) = πx ) cos x. sin x + cos x x 65. Show that d dx [xn J n (x)] = x n J n 1 (x). 66. Show that d [ x n J n (x) ] = x n J n+1 (x). dx 67. Show that J n (x) = 1 2 (J n 1(x) J n+1 (x)). ). 68. Show that J n (x) = 1 4 (J n 2(x) 2J n (x) + J n+2 (x)).