SPS Mathematical Methods

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1 SPS Mathematical Methods Assignment No. 2 Deadline: 11th March 2015, before 4:45 p.m. INSTRUCTIONS: Answer the following questions. Check our answer for odd number questions at the back of the tetbook. 2.4 Eercises 11, 15, 17, 19, Eercises 7, 9, Eercises 9, 11, 13, 15, 17 19, 21, 23, 25, 27 29, 31, 33, 35, Review Problems 31, 33, 35, 37, 39 1

2 Section 2.4 Eact Equations 61 Remark. Since we can use either procedure for finding FA, B, it ma be worthwhile to consider each of the integrals MA, B and NA, B. If one is easier to evaluate than the other, this would be sufficient reason for us to use one method over the other. [The skeptical reader should tr solving equation (15) b first integrating MA, B.] Eample 4 Solution Show that (16) A ϩ 3 3 sin B ϩ A 4 cos B ϭ 0 is not eact but that multipling this equation b the factor this fact to solve (16). In equation (16), M ϭ ϩ 3 3 sin and N ϭ 4 cos. Because Ϫ1 ields an eact equation. Use equation (16) is not eact. When we multipl (16) b the factor Ϫ1, we obtain (17) 0M 0 ϭ 33 cos [ 4 3 cos ϭ 0N 0, A1 ϩ 3 2 sin B ϩ A 3 cos B ϭ 0. For this new equation, M ϭ 1 ϩ 3 2 sin and N ϭ 3 cos. If we test for eactness, we now find that 0M 0 ϭ 32 cos ϭ 0N 0, and hence (17) is eact. Upon solving (17), we find that the solution is given implicitl b ϩ 3 sin ϭ C. Since equations (16) and (17) differ onl b a factor of, then an solution to one will be a solution for the other whenever 0. Hence the solution to equation (16) is given implicitl b ϩ 3 sin ϭ C. In Section 2.5 we discuss methods for finding factors that, like ineact equations into eact equations. Ϫ1 in Eample 4, change 2.4 EXERCISES In Problems 1 8, classif the equation as separable, linear, eact, or none of these. Notice that some equations ma have more than one classification A 10 / 3 Ϫ 2B ϩ ϭ 0 A 2 ϩ 4 cos B Ϫ 3 ϭ Ϫ2 Ϫ 2 ϩ A3 ϩ 2 Ϫ 2 B ϭ 0 4. Ae ϩ 2B ϩ Ae Ϫ 2B ϭ 0 5. ϩ ϭ In Problems 9 20, determine whether the equation is eact. If it is, then solve it ϩ A2 ϩ cos B ϭ ϩ cos AB 4 ϩ 3 cos AB Ϫ 2 4 ϭ 0 u dr ϩ A3r Ϫ u Ϫ 1B du ϭ 0 A2 ϩ B ϩ A Ϫ 2B ϭ A2 ϩ 3B ϩ A 2 Ϫ 1B ϭ 0

3 62 Chapter 2 First-Order Differential Equations 11. Acos cos ϩ 2B Ϫ Asin sin ϩ 2B ϭ In Problems 21 26, solve the initial value problem Ae sin Ϫ 3 2 B ϩ Ae cos ϩ Ϫ2 / 3 /3B ϭ 0 At/B ϩ A1 ϩ ln B dt ϭ 0 e t A Ϫ tb dt ϩ A1 ϩ e t B ϭ 0 cos u dr Ϫ Ar sin u Ϫ e u B du ϭ 0 A e Ϫ 1/B ϩ Ae ϩ / 2 B ϭ 0 A1/B Ϫ A3 Ϫ / 2 B ϭ ϩ 2 Ϫ cosa ϩ B 4 ϩ 3 2 Ϫ cosa ϩ B Ϫ e 4 ϭ 0 a2 ϩ 2 c ϩ cosab d 2 21 Ϫ A1/ ϩ 2 2 B ϩ A2 2 Ϫ cos B ϭ 0, A e Ϫ 1/B ϩ Ae ϩ / 2 B ϭ 0, Ae t ϩ te t B dt ϩ Ate t ϩ 2B ϭ 0, Ae t ϩ 1B dt ϩ Ae t Ϫ 1B ϭ 0, A1B ϭ 1 A 2 sin B ϩ A1/ Ϫ /B ϭ 0, ApB ϭ 1 Atan Ϫ 2B ϩ A sec 2 ϩ 1/B ϭ 0, 27. For each of the following equations, find the most general function MA, B so that the equation is eact. (a) (b) 1 ϩ 2 2b ϩ a 1 ϩ 2 Ϫ 2b ϭ 0 2 ϩ 3 cos AB Ϫ Ϫ1 / 3 4 ϭ 0 A1B ϭ p A1B ϭ 1 A0B ϭ 1 M A, B ϩ Asec 2 Ϫ /B ϭ 0 M A, B ϩ Asin cos Ϫ Ϫ e Ϫ B ϭ For each of the following equations, find the most general function NA, B so that the equation is eact. (a) 3 cos AB ϩ e 4 ϩ N A, B ϭ 0 (b) Ae Ϫ 4 3 ϩ 2B ϩ N A, B ϭ Consider the equation A 2 ϩ 2B Ϫ 2 ϭ 0. A0B ϭ Ϫ1 (a) Show that this equation is not eact. (b) Show that multipling both sides of the equation b Ϫ2 ields a new equation that is eact. (c) Use the solution of the resulting eact equation to solve the original equation. (d) Were an solutions lost in the process? 30. Consider the equation (a) Show that the equation is not eact. (b) Multipl the equation b n m and determine values for n and m that make the resulting equation eact. (c) Use the solution of the resulting eact equation to solve the original equation. 31. Argue that in the proof of Theorem 2 the function g can be taken as which can be epressed as This leads ultimatel to the representation (18) A5 2 ϩ ϩ 4 2 B ϩ A2 3 ϩ 3 4 ϩ 3 2 B ϭ 0. gab ϭ Ύ NA, tb dt Ϫ 0 Ύ c 0 0t Ύ gab ϭ Ύ 0 NA, tb dt Ϫ Ύ ϩ Ύ FA, B ϭ Ύ 0 MAs, 0 B ds. 0 MAs, B ds 0 NA, tb dt ϩ Ύ Evaluate this formula directl with to rework (a) Eample 1. (b) Eample 2. (c) Eample 3. 0 MAs, 0 B ds. 0 ϭ 0, 0 ϭ Orthogonal Trajectories. A geometric problem occurring often in engineering is that of finding a famil of curves (orthogonal trajectories) that intersects a given famil of curves orthogonall at each point. For eample, we ma be given the lines of force of an electric field and want to find the equation 0 0 MAs, tb ds d dt,

4 Section 2.4 Eact Equations 63 for the equipotential curves. Consider the famil of curves described b FA, B ϭ k, where k is a parameter. Recall from the discussion of equation (2) that for each curve in the famil, the slope is given b F ~ F. (a) Recall that the slope of a curve that is orthogonal (perpendicular) to a given curve is just the negative reciprocal of the slope of the given curve. Using this fact, show that the curves orthogonal to the famil FA, B ϭ k satisf the differential equation F F A, B A, B 0. Figure 2.11 Families of orthogonal hperbolas (b) Using the preceding differential equation, show that the orthogonal trajectories to the famil of circles 2 ϩ 2 ϭ k are just straight lines through the origin (see Figure 2.10). Figure 2.10 Orthogonal trajectories for concentric circles are lines through the center (c) Show that the orthogonal trajectories to the famil of hperbolas ϭ k are the hperbolas 2 Ϫ 2 ϭ k (see Figure 2.11). 33. Use the method in Problem 32 to find the orthogonal trajectories for each of the given families of curves, where k is a parameter. (a) (b) (c) (d) 2 2 ϩ 2 ϭ k ϭ k 4 ϭ e k 2 ϭ k [Hint: First epress the famil in the form F(, ) ϭ k.] 34. Use the method described in Problem 32 to show that the orthogonal trajectories to the famil of curves 2 ϩ 2 ϭ k, k a parameter, satisf A2 Ϫ1 B ϩ A 2 Ϫ2 Ϫ 1B ϭ 0. Find the orthogonal trajectories b solving the above equation. Sketch the famil of curves, along with their orthogonal trajectories. [Hint: Tr multipling the equation b m n as in Problem 30.] 35. Using condition (5), show that the right-hand side of (10) is independent of b showing that its partial derivative with respect to is zero. [Hint: Since the partial derivatives of M are continuous, Leibniz s theorem allows ou to interchange the operations of integration and differentiation.] 36. Verif that FA, B as defined b (9) and (10) satisfies conditions (4).

5 Section 2.5 Special Integrating Factors 67 Because (12) is not eact, we compute We obtain a function of onl, so an integrating factor for (12) is given b formula (8). That is, When we multipl (12) b m ϭ Ϫ2, we get the eact equation Solving this equation, we ultimatel derive the implicit solution (13) 0M/0 Ϫ 0N/0 1 Ϫ A2 Ϫ 1B 2A1 Ϫ B ϭ N 2 ϭ Ϫ ϪA1 Ϫ B ϭ Ϫ2 mab ϭ ep a Ύ Ϫ2 b ϭ Ϫ2. A2 ϩ Ϫ2 B ϩ A Ϫ Ϫ1 B ϭ 0. 2 Ϫ Ϫ1 ϩ 2 2 ϭ C. Notice that the solution ϵ 0 was lost in multipling b m ϭ Ϫ2. Hence, (13) and ϵ 0 are solutions to equation (12). There are man differential equations that are not covered b Theorem 3 but for which an integrating factor nevertheless eists. The major difficult, however, is in finding an eplicit formula for these integrating factors, which in general will depend on both and EXERCISES In Problems 1 6, identif the equation as separable, linear, eact, or having an integrating factor that is a function of either alone or alone A2 ϩ Ϫ1 B ϩ A Ϫ 1B ϭ 0 A2 3 ϩ 2 2 B ϩ A3 2 ϩ 2B ϭ 0 A2 ϩ B ϩ A Ϫ 2B ϭ 0 A 2 ϩ 2B Ϫ 2 ϭ 0 A 2 sin ϩ 4B ϩ ϭ 0 A2 2 Ϫ B ϩ ϭ 0 In Problems 7 12, solve the equation A2B ϩ A 2 Ϫ 3 2 B ϭ 0 A3 2 ϩ B ϩ A 2 Ϫ B ϭ 0 A 4 Ϫ ϩ B Ϫ ϭ 0 A2 2 ϩ 2 ϩ 4 2 B ϩ A2 ϩ B ϭ 0 A 2 ϩ 2B Ϫ 2 ϭ A2 3 ϩ 1B ϩ A3 2 2 Ϫ Ϫ1 B ϭ 0 In Problems 13 and 14, find an integrating factor of the form n m and solve the equation. 13. A2 2 Ϫ 6B ϩ A3 Ϫ 4 2 B ϭ A12 ϩ 5B ϩ A6 Ϫ1 ϩ 3 2 B ϭ (a) Show that if A0N/ 0 Ϫ 0M/ 0B / AM Ϫ NB depends onl on the product, that is, 0N/ 0 Ϫ 0M/ 0 ϭ HAB, M Ϫ N then the equation MA, B ϩ NA, B ϭ 0 has an integrating factor of the form mab. Give the general formula for mab. (b) Use our answer to part (a) to find an implicit solution to (3 ϩ 2 2 ) ϩ ( ϩ 2 2 ) ϭ 0, satisfing the initial condition (1) ϭ (a) Prove that M ϩ N ϭ 0 has an integrating factor that depends onl on the sum ϩ if and onl if the epression 0N/ 0 Ϫ 0M/ 0 M Ϫ N depends onl on ϩ. (b) Use part (a) to solve the equation (3 ϩ ϩ ) ϩ (3 ϩ ϩ ) ϭ 0.

6 68 Chapter 2 First-Order Differential Equations 17. (a) Find a condition on M and N that is necessar and sufficient for M ϩ N ϭ 0 to have an integrating factor that depends onl on the product 2. (b) Use part (a) to solve the equation (2 ϩ 2 ϩ 2 3 ϩ ) ϩ (2 ϩ 4 ϩ 2 3 ) ϭ If MA, B ϩ NA, B ϵ 0, find the solution to the equation MA, B ϩ NA, B ϭ Fluid Flow. The streamlines associated with a certain fluid flow are represented b the famil of curves ϭ Ϫ 1 ϩ ke Ϫ. The velocit potentials of the flow are just the orthogonal trajectories of this famil. (a) Use the method described in Problem 32 of Eercises 2.4 to show that the velocit potentials satisf ϩ A Ϫ B ϭ 0. [Hint: First epress the famil ϭ Ϫ 1 ϩ ke Ϫ in the form FA, B ϭ k.] (b) Find the velocit potentials b solving the equation obtained in part (a). 20. Verif that when the linear differential equation 3PAB Ϫ QAB4 ϩ ϭ 0 is multiplied b mab ϭ e PAB, the result is eact. 2.6 SUBSTITUTIONS AND TRANSFORMATIONS When the equation MA, B ϩ NA, B ϭ 0 is not a separable, eact, or linear equation, it ma still be possible to transform it into one that we know how to solve. This was in fact our approach in Section 2.5, where we used an integrating factor to transform our original equation into an eact equation. In this section we stu four tpes of equations that can be transformed into either a separable or linear equation b means of a suitable substitution or transformation. Substitution Procedure (a) Identif the tpe of equation and determine the appropriate substitution or transformation. (b) Rewrite the original equation in terms of new variables. (c) Solve the transformed equation. (d) Epress the solution in terms of the original variables. Homogeneous Equations Homogeneous Equation Definition 4. If the right-hand side of the equation (1) ϭ f A, B can be epressed as a function of the ratio / alone, then we sa the equation is homogeneous.

7 74 Chapter 2 First-Order Differential Equations The last equation is homogeneous, so we let z ϭ /u. Then /du ϭ z ϩ uadz/dub, and, substituting for /u, we obtain dz z ϩ u du ϭ 3 Ϫ z 1 ϩ z. Separating variables gives Ύ z ϩ 1 z 2 ϩ 2z Ϫ 3 dz ϭ Ϫ Ύ 1 u du, 1 2 ln 0 z2 ϩ 2z Ϫ 3 0 ϭ Ϫln 0 u 0 ϩ C 1, from which it follows that z 2 ϩ 2z Ϫ 3 ϭ Cu Ϫ2. When we substitute back in for z, u, and, we find A/uB 2 ϩ 2A/uB Ϫ 3 ϭ Cu Ϫ2, 2 ϩ 2u Ϫ 3u 2 ϭ C, A ϩ 3B 2 ϩ 2A Ϫ 1B A ϩ 3B Ϫ 3A Ϫ 1B 2 ϭ C. This last equation gives an implicit solution to (16). 2.6 EXERCISES In Problems 1 8, identif (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form ϭ GAa ϩ bb. 1. 2t ϩ At 2 Ϫ 2 B dt ϭ 0 2. A Ϫ 4 Ϫ 1B 2 Ϫ ϭ 0 3. / ϩ / ϭ At ϩ ϩ 2B ϩ A3t Ϫ Ϫ 6B dt ϭ Use the method discussed under Homogeneous Equations to solve Problems A ϩ 2 B Ϫ 2 ϭ A3 2 Ϫ 2 B ϩ A Ϫ 3 Ϫ1 B ϭ A 2 Ϫ B ϩ 2 ϭ A 2 ϩ 2 B ϩ 2 ϭ u Ϫ du ϭ 2u du A e Ϫ2 ϩ 3 B Ϫ e Ϫ2 ϭ 0 cosa ϩ B ϭ sin A ϩ B A 3 Ϫ u 2 B du ϩ 2u 2 ϭ 0 dt ϭ 2 ϩ t2t 2 ϩ 2 t du ϭ u sec A /ub ϩ u ϭ 2 Ϫ 2 3 Use the method discussed under Equations of the Form / ϭ GAa ϩ bb to solve Problems / ϭ 2 ϩ Ϫ / ϭ A ϩϩ2b2 19. / ϭ A Ϫ ϩ 5B / ϭ sina Ϫ B Use the method discussed under Bernoulli Equations to solve Problems ϩ ϭ Ϫ ϭ e2 3 ϭ 2 Ϫ 2 2 ϩ Ϫ 2 ϭ 5 A Ϫ / 2B dt ϩ t3 ϩ t ϭ 0 dr du ϭ r2 ϩ 2ru u 2 Aln Ϫ ln ϩ 1B ϭ ϩ ϭ e Ϫ2 ϩ 3 ϩ ϭ 0

8 Section 2.6 Substitutions and Transformations 75 Use the method discussed under Equations with Linear Coefficients to solve Problems AϪ3 ϩ Ϫ 1B ϩ A ϩ ϩ 3B ϭ A ϩ Ϫ 1B ϩ A Ϫ Ϫ 5B ϭ A2 Ϫ B ϩ A4 ϩ Ϫ 3B ϭ A2 ϩ ϩ 4B ϩ A Ϫ 2 Ϫ 2B ϭ 0 In Problems 33 40, solve the equation given in: 33. Problem Problem Problem Problem Problem Problem Problem Problem Use the substitution ϭ Ϫ ϩ 2 to solve equation (8). 42. Use the substitution ϭ 2 to solve ϭ 2 ϩ cosa / 2 B. 43. (a) Show that the equation / ϭ f A, B is homogeneous if and onl if f At, tb ϭ f A, B. [Hint: Let t ϭ 1/.] (b) A function HA, B is called homogeneous of order n if HAt, tb ϭ t n HA, B. Show that the equation MA, B ϩ NA, B ϭ 0 is homogeneous if MA, B and NA, B are both homogeneous of the same order. 44. Show that equation (13) reduces to an equation of the form ϭ GAa ϩ bb, when a 1 b 2 ϭ a 2 b 1. [Hint: If a 1 b 2 ϭ a 2 b 1, then a 2/a 1 ϭ b 2/b 1 ϭ k, so that a 2 ϭ ka 1 and b 2 ϭ kb 1.] 45. Coupled Equations. In analzing coupled equations of the form ϭ a ϩ b, dt ϭ a ϩ b, dt where a, b, a, and b are constants, we ma wish to determine the relationship between and rather than the individual solutions AtB, AtB. For this purpose, divide the first equation b the second to obtain (17) This new equation is homogeneous, so we can solve it via the substitution ϭ /. We refer to the solutions of (17) as integral curves. Determine the integral curves for the sstem 46. Magnetic Field Lines. As described in Problem 20 of Eercises 1.3, the magnetic field lines of a dipole satisf Solve this equation and sketch several of these lines. 47. Riccati Equation. An equation of the form (18) a ϩ b ϭ a ϩ b. dt ϭ Ϫ4 Ϫ, dt ϭ 2 Ϫ. ϭ Ϫ 2. PAB 2 QAB RAB is called a generalized Riccati equation. (a) If one solution sa, uab of (18) is known, show that the substitution ϭ u ϩ 1/ reduces (18) to a linear equation in. (b) Given that uab ϭ is a solution to ϭ 3 A Ϫ B 2 ϩ, use the result of part (a) to find all the other solutions to this equation. (The particular solution uab ϭ can be found b inspection or b using a Talor series method; see Section 8.1.) Historical Footnote: Count Jacopo Riccati studied a particular case of this equation in 1724 during his investigation of curves whose radii of curvature depend onl on the variable and not the variable.

9 Review Problems 77 REVIEW PROBLEMS In Problems 1 30, solve the equation Ϫ 4 ϭ 322 ϭ eϩ Ϫ 1 3. A 2 Ϫ 2 Ϫ3 B ϩ A2 Ϫ 3 2 B ϭ 0 4. ϩ 3 ϭ 2 Ϫ 4 ϩ sin AB ϩ cos AB4 ϩ 31 ϩ 2 cosab4 ϭ Ϫ A1 Ϫ 2 B ϭ 0 t 3 2 dt ϩ t 4 Ϫ6 ϭ 0 ϩ 2 ϭ 22 2 A 2 ϩ 2 B ϩ 3 ϭ ϩ A1 ϩ 2 ϩ 2 ϩ 2 B Ϫ1 4 ϩ 3 Ϫ1 / 2 ϩ A1 ϩ 2 ϩ 2 ϩ 2 B Ϫ1 4 ϭ 0 dt ϭ 1 ϩ cos2 At Ϫ B A 3 ϩ 4e B ϩ A2e ϩ 3 2 B ϭ 0 Ϫ ϭ 2 sin 2 dt Ϫ t Ϫ 1 ϭ t2 ϩ 2 ϭ 2 Ϫ 22 Ϫ ϩ 3 ϩ tan ϩ sin ϭ 0 ϩ 2 ϭ 2 du 18. ϭ A2 ϩ Ϫ 1B2 19. A 2 Ϫ 3 2 B ϩ 2 ϭ du ϩ u ϭ Ϫ4uϪ2 21. A Ϫ 2 Ϫ 1B ϩ A ϩ Ϫ 4B ϭ A2 Ϫ 2 Ϫ 8B ϩ A Ϫ 3 Ϫ 6B ϭ A Ϫ B ϩ A ϩ B ϭ In Problems 31 40, solve the initial value problem. 31. ( 3 Ϫ ) ϩ ϭ 0, (1) ϭ A2/ ϩ cos B ϩ A2/ ϩ sin B ϭ 0 A Ϫ Ϫ 2B ϩ A Ϫ ϩ 4B ϭ 0 ϩ ϭ 0 A3 Ϫ Ϫ 5B ϩ A Ϫ ϩ 1B ϭ 0 ϭ Ϫ Ϫ 1 ϩ ϩ 5 A4 3 Ϫ 9 2 ϩ 4 2 B ϩ A3 2 2 Ϫ 6 ϩ 2 2 B ϭ 0 ϭ A ϩ ϩ 1B2 Ϫ A ϩ Ϫ 1B 2 ϭ a ϩ b, A1B ϭ Ϫ4 33. At ϩ ϩ 3B dt ϩ ϭ 0, A0B ϭ Ϫ 2 ϭ 2 cos, ApB ϭ A2 2 ϩ 4 2 B Ϫ ϭ 0, A1B ϭ Ϫ2 3 2 cosa2 ϩ B Ϫ 2 4 ϩ 3 cosa2 ϩ B ϩ e 4 ϭ 0, A1B ϭ 0 A2 Ϫ B ϩ A ϩ Ϫ 3B ϭ 0, A0B ϭ 2 2 ϩ A 2 ϩ 4B ϭ 0, A0B ϭ 4 Ϫ 2 ϭ Ϫ1 Ϫ1, A1B ϭ Ϫ 4 ϭ 22, A0B ϭ Ϫ4 41. Epress the solution to the following initial value problem using a definite integral: dt ϭ 1 1 ϩ t 2 Ϫ, A2B ϭ 3. Then use our epression and numerical integration to estimate (3) to four decimal places.