366 MA Spring 2011 Assignments
For Wednesday 1/12: Read 1.21.3. Do: p. 25: 7, 9, 16 p. 360, The answer to Exercise 8 is given in the back of the text. Substitute the given functions x1 and x2 into the the system to show that they do solve the system. Show that they also satisfy the desired initial conditions stated in the exercise. For Friday 1/14: Read p. 31-39 Do:Exercises : p. 25: 13, 14 p. 360, The answer to Exercise 10 is given in the back of the text. Substitute the given functions x1 and x2 into the the system to show that they do solve the system. Show that they also satisfy the desired initial conditions stated in the exercise. Exercises p. 39: 1(use deld), 4(use deld), 13,15, 16 Note: Note: In the deld problems to determine the behavior as t goes to innity you need to also draw several of the solutions as the text did in gure 2.1.4 on p. 38 and describe what you see. In part (c) you are asked to prove that your observation is correct. For Wednesday 1/19: Read p. 21, rst paragraph, p. 42-47 p. 39: 14,19, 24 (use deld) p. 24: The dierential equation in exercise 4 is not linear. Change one single number to make the equation linear. Do not just change one of the coecients to zero! p. 47, 1, 7, 10 (use deld), 23. (Hint: Set y = 0 and solve for x. You dont need to compute y to do this! How can you tell that it is a minimum and not a maximum or a point of inection?) 1
For Friday 1/21: In all of these problems, do not do the computer part! Read Example 1, p. 52. Then do Exercise 4, p. 60. Only nd the concentration prior to the point of overow. Do not do the parts described in the nal two sentences of the problem. p. 48, 22 (This is similar to 23, but now you want y = .),24 (Prove that your value really is a maximum.), 30 (a)-(e) p. 62, 16. For Wednesday 1/26 In all of these problems, do not do the computer part! p. 50, 31(a), (b) p.202, 1 (See the discussion relating to Equation (12) pn p. 195. You will need the technique used in this problem for 5(c) below.) p. 60, 5, 20 Note in some versions of the text the rate in and rate out are given as 3 gal/min. Please use 2 gal/min instead. For Friday , 1/28 p. 100 1, 2, 3, 21 In all of these problems, do not do the computer part! p. 50, 32(a), (b) p. 60, 21(a), (b) dv dy in terms of v and . p. 77, 27(b)Hint You need to express dx dx 1. At time t = 0, a tank of water contains 500gal of water with 50g of salt dissolved in it. Water ows into the tank at 30gal/hr and the well stirred mixture ows out at the same rate. Assume that the concentration of the salt coming in at time t is is 10et lb/gal. Write a dierential equation and initial value condition which could be solved to nd a formula for the amount Q(t) of salt in the tank. You need not solve the equation. 2. Let the data be exactly the same as in the preceding problem except now water ows out of the tank at 40gal/hr. Write a dierential equation and initial value condition which could be solved to nd a formula for the amount Q(t) of salt in the tank at any time prior to the time 2
the tank become empty. You need not solve the equation. For Wednesday 2/2 p. 77: 28 p. 100: 7, 11, 15, 22 1. An object of mass 1 kg. is thrown upward with an initial velocity of 20 m/s from a building 30 m high. There is a force of resistance of 9.8v 2 where the velocity v is measured in m/s. (1) Find a formula for the velocity v (t) for the object when it is rising. Then (2) Find the time to at which it reaches its maximum height. Finally, (3) Find a formula for v (t) for t > to which is valid until the object hits the ground. You may leave this answer in implicit form. Ans: Going up: v (t) = tan(1.52 9.8t), to = 0.155, Going down: 1+v = e19.6(t0.155) . 1v Remark. The numbers in this problem are not meant to be realistic. They are chosen to make the calculations easier. 2. The statement of the extra Exercises 1 and 2 from the assignment for Friday , 1/28, have been corrected. (concentration was changed to amount.) Redo these exercises with the corrected statement. 3. For the following dierential equation (a) nd the general solution (b) nd a value of yo for which the initial value problem y (0) = yo has two solutions. (c) Explain, using the Existence and Uniqueness Theorem (Theorem 2.8.1 on p.112), why there is only one possible choice of yo in (b) (d) Solve the initial value problem y (0) = 1. There is only one solution! Hint Completing the square helps in solving for y in terms of x. (3 + 2y )y = 3x2 1 4. For the following dierential equation (a) nd the general solution (b) Find a value of to for which the initial value problem y (to ) = 1 has no solution. (c) Explain, using the Existence and Uniqueness Theorem (Theorem 2.8.1 on p.112), why there is only one possible choice of to in (b) ty + 2y = t2 et 3
5. For the following dierential equation (a) nd the general solution (b) Find a value of yo for which the initial value problem y (3) = yo has more than one solution. (c) Explain, using the Existence and Uniqueness Theorem (Theorem 2.8.1 on p.112), why there is only one possible choice of yo in (b) y = (2y 3)3/5 (x 4)2
For Wednesday, 2/16 Read 2.5, p. 78-81, 3.1, p. 137-143, 3.3, p.157-163 Do: p.89, 8, 10 p. 144, 5,6,7, 9, 12 (Do not graph), p. 163, 1, 3, 7, 18 For Friday, 2/18/09 In all of these problems, do not do the computer part and do not graph p. 163, 11, 19, 34(a), (b) p. 144, 17,21, p. 171, 2, 6 1. Consider the dierential equation y (4) + 4y + 3y 4y 4y = 0. Let L(y ) be the corresponding linear operator on functions. (a) Show that L(ert ) = (r2 1)(r + 2)2 ert . (b) Use the fact that L to show that L tert = 2r(r + 2)2 + 2(r2 1)(r + 2) + t(r2 1)(r + 2)2 ert . (c) Use the above formula to show that L (te2t ) = 0. 4 y r = L(y ) r (1)
(d) Verify by direct substitution into equation (1) that L (te2t ) = 0. For Wednesday, 2/23/09 In all of these problems, do not do the computer part and do not graph Read 3.5, p. 174-183, 3.1: p. 144, 23 3.3: p. 165, 36 3.4 p. 171, 12 3.5 p. 183, 1, 2,3,5 For Friday, 2/25 In all of these problems, do not do the computer part and do not graph Read The section on variation of parameters. (3.6) 3.5: p. 183, 8, 29 3.6: p. 189, 5, 17 Note: In 17 you must divide the dierential equation by x2 before applying the technique. In the problems on p. 189 DO NOT use the texts formulas (26) and (27). Instead, explicitly write the equations u1 y 1 + u2 y 2 = 0 u1 y1 + u2 y2 = g (t) as I did in class where y1 and y2 and g are EXPLICIT functions. (The yi are the fundamental set of solutions and the g is the function on the right side of the dierential equation.) Then express this system in terms of matrices and use Cramers rule to nd u1 and u2 . Finally, integrate to nd u1 , u2 , and the general solution to the equation, which is y = u1 y1 + u2 y2 . 5
For Wednesday, 3/2 In all of these problems, do not do the computer part and do not graph 3.1 p. 144, 24 3.3 p. 165, 39 3.5 p. 183, 30 3.7 p. 202, 5 Note: The information in the rst sentence of the statement of the problem is needed to nd the spring constant k . See Example 1, p. 194. p. 202, 9: Do only the part up to Plot u versus t. (You may use a 22 + (5/ 6)2 e10t and computer.) Also plot on the same graph y = y= 22 + (5/ 6)2 e10t .
3.6 p. 189, 10 DO NOT use the texts formulas (26) and (27). Instead, explicitly write the equations u1 y1 + u2 y2 = 0 u1 y1 + u2 y2 = g (t) as I did in class where y1 and y2 and g are EXPLICIT functions. (The yi are the fundamental set of solutions and the g is the function on the right side of the dierential equation.) Then express this system in terms of matrices and use Cramers rule to nd u1 and u2 . Finally, integrate to nd u1 , u2 , and the general solution to the equation, which is y = u1 y 1 + u2 y 2 . For Friday, 3/4 3.6: p. 189, 11 (DO NOT use the texts formulas (26) and (27).) 3.7: p. 202 6, 10: Do only the part up to Plot u versus t. (You may use a computer.) 13 Additional problem: 1. In Exercise 5 on p. 202, for which coecients of damping would the motion be over damped? Critically damped? Under damped? 6
p. 215, 1 (See Fridays class notes), 5, 6, 7(a), 8(a), (b) (The steady state solution is the part that is not multiplied by eat for some a > 0. The transient part is the part that is multiplied by eat .),9, 15, 17 For Wednesday, 3/30 p. 259: 2 (a),(b) (See Example 2, p. 255) The solution y1 is found by setting a0 = 1 and a1 = 0 in the recursion relation. The solution y2 is found by setting a0 = 0 and a1 = 1 in the recursion relation. For Friday, 4/1 p. 259: 7(a), (b), 17(a), (b) 1. Repeat the instructions for 7(a), (b) on p. 259 for the equation x2 y + (x3 2)y = 0. For Wednesday, 4/6 p. 259: Re-do Exercise 7(a), (b) with xo = 1, p. 259: 6 (a), (b) 10(a), (b) 1. Find a formula for the Wronskian W (x) for the solutions y1 and y2 in Exercise 6, p. 259. Use following the theorem: Theorem 1. Suppose that y1 and y2 satisfy L(y1 ) = L(y2 ) = 0 where L(y ) = y + py + qy. Then the Wronskian W of y1 and y1 satises W + pW = 0. 2. Determine the largest r such that f (x) has a series expansion
f ( x) =
0
an xn
which is valid for all x, |x| < r for the following functions f (x). (See Examples 1, 2 on p. 263-264.) You are not asked to nd the expansion. 7
. Ans: r = 2 3. x3 (b) f (x) = x2 +4x+8 . Ans: r = 2 2. (a) f (x) =
x3 x2 x+12
p. 265: 9, Problem 6. Use the following theorem which follows from Theorem 5.3.1 on p. 262 of the text: Theorem 2. Suppose that y satises y + p(x)y + q (x)y = 0 where p(x) has a power series expansion that converges for |x| < r1 and and q (x) has a power series expansion that converges for |x| < r2 . Then y (x) has a power series expansion which converges at least for all |x| < min{r1 , r2 }. For Friday, 4/8 p. 265, 7. Note: x3 + 1 = (x + 1)(x2 x + 1). 1. Let L(y ) = y + py + qy where p and q are functions. (a) Suppose that y2 = uy1 where u and y1 are functions. Show that L(uy1 ) = u y1 + (2y1 + py1 )u + uL(y1 ). (I essentially did this Wednesday. Look at the notes from the last part of class.) (b) Suppose that L(y1 ) = 0. Let u be a function and set v = u . Show that L(uy1 ) = 0 if and only if v + (2(ln y1 ) + p)v = 0. (c) Show that the function v from part 1b is given by the formula
v = Cy1 2 e p dx
.
(2)
(d) Use the formula just derived to do problem 36 on p. 174 of the text. Specically, y2 = y1 u where u = v and v is computed from formula (2). 8
For Wednesday, 4/13 At the end of Mondays class I started to state the stated the following denition: Denition 1. We say that a dierential equation has regular singularities at x = 0 if it is equivalent to an equation of the form x2 p(x)y + xq (x)y + s(x)y = 0 (3)
where p, q , and s are analytic at x = 0(i.e. they have power series expansions in powers of x which converge in an interval containing x = 0.) and where p(0) = 0. In this case the equation x2 p(0)y + xq (0)y + s(0)y = 0 is referred to as the approximating Euler equation. The equation p(0)r(r 1) + q (0)r + s(0) = 0 is the indicial equation and its roots are the exponents at the singularity. Note that if we set P (x) = x2 p(x), Q(x) = xq (x) and R(x) = r(x) then q (x) xQ(x) = lim x0 p(x) x0 P (x) q (0) = p(0) 2 x R(x) r ( x) lim = lim x0 P (x) x0 p(x) r(0) = p(0) lim This leads to the Books denition: Denition 2 (Book). We say that a dierential equation has regular singularities at x = 0 if it is equivalent to an equation of the form P (x)y + Q(x)y + R(x)y = 0 9 (4)
where P , Q, and R are analytic at x = 0(i.e. they have power series expansions in powers of x which converge in an interval containing x = 0.) and where the limits xQ(x) po = lim x0 P (x) x 2 R ( x) qo = lim x0 P (x) both exist. In this case the equation x2 y + xpo y + qo y = 0 is referred to as the approximating Euler equation. The equation r(r 1) + po r + qo = 0 is the indicial equation and its roots are the exponents at the singularity. It is a theorem that both denitions are equivalentany equation that has regular singularities under the books denition has regular singularities under my denition and vice versa. The books denition gives a slightly dierent indicial equation than my denition. From the above discussion po = q (0) , p(0) qo = r(0) . p(0)
Hence the books indicial equation is r(r 1) + q (0) r(0) r+ =0 p(0) p(0)
which is equivalent with mine. I prefer my denition because you dont need limits to tell if an equation has regular singularities: if it has an approximating Euler equation, then it has regular singularities. For example the equation (x5 + 2x3 + x2 )y + (3x2 + 7x)y + (x2 + 3x + 5)y = 0 may be written x2 (x3 + 2x + 1)y + x(3x + 7)y + (x2 + 3x + 5)y = 0. 10
Hence in (3) p(x) = x3 + 2x + 1, q (x) = 3x + 7, and r(x) = x2 + 3x + 5.
Since p(x), q (x), and r(x) are polynomials they are analytic at x = 0. Also p(0) = 0. Hence the equation has regular singularities at x = 0. From formula (4) above the approximating Euler equation is x2 y + 7xy + 5y = 0. The equation (x5 + 2x3 + x2 )y + (3x2 + 7)y + (x2 + 3x + 5)y = 0 does not have regular singularities. To see this we rst write the equation as 7 x2 (x3 + 2x + 1)y + x(3x + )y + (x2 + 3x + 5)y = 0. x
7 We cannot let q (x) = 3x + x since this is not dened at x = 0. To avoid this we multiply both sides of our equation by x writing our equation as
x2 (x4 + 2x2 + x)y + x(3x2 + 7)y + (x3 + 3x2 + 5x)y = 0. But now p(x) = x4 + 2x2 + x so p(0) = 0, which is not allowed. The Assignment: 1. If we write Problem 8 on p. 282 as in equation (3) above, what are p(x), q (x), and r(x)? What is the approximating Euler equation? 2. Use my denition of regular singularities to do parts (a) and (b) of Problem 8 on p. 282. 3. Use the books denition of regular singularities to do parts (a) and (b) of Problem 8 on p. 282. 4. Do parts (c) and (d) of Problem 8 on p. 282. 5. Use formula (2) from last Fridays assignment to do problem 37 on p. 174 of the text. 11
6. Let L(y ) = y + py + qy where p and q are functions. (a) Suppose that y2 = uy1 where u and y1 are functions. It was shown in last Fridays homework that L(uy1 ) = u y1 + (2y1 + py1 )u + uL(y1 ). Suppose that L(y1 ) = 0. Let u and r be functions and set v = u . Show that L(uy1 ) = r if and only if
v + (2(ln y1 ) + p)v = y1 1 r.
(b) Show that the function v is given by the formula
v = y1 2 e p dx
r y1 e
p dx
dx.
(5)
(c) Use the formula just derived to do problem 17 on p. 189 of the text. This demonstrates that we can use reduction of order as a replacement for variation of parameters.
For Friday, 4/15 I have posted copies of the exercises from the remaining sections of the text that might be covered in the le Some problems from the text. The solutions are contained in the le Solutions to the problems from the text. If your text diers from the one I am using, you should do these problems. I have also (nally) posted solutions to Test 2. 1. Use my denition of regular singularities to do parts (a) and (b) of Problem 5 on p. 282. 2. Use the books denition of regular singularities to do parts (a) and (b) of Problem 5 on p. 282. 3. Do parts (c) and (d) of Problem 2 on p. 282. 12
4. For the dierential equation in Exercise 5 on p. 282, give the approximating Euler equation. Use it to nd the indicial equation and its roots. Use Theorem 5.6.1 on p. 289 to describe the expected form of the solutions y1 and y2 . Do not nd the coecients! This is meant to be a short exercise. 5. Repeat Exercise 4 for Exercise 9 on p. 282. 6. Use formula (2) from Wednesdays assignment to do problem 35 on p. 174 of the text: Find a second solution to ty y + 4t3 y = 0, y1 (t) = sin(t2 ). 7. Use formula (2) from last Fridays assignment to do problem 7 on p. 189 of the text: Find a particular solution to y + 4y + 4y = t2 e2t .
p. 359: 1(See Example 1, p. 357), 3, 7(a). By solve for x2 , they mean write x2 = x1 + 2x1 . For Wednesday, 4/10 Read p.390-398. (See Example 5 on p. 381.) p. 398, 2(a), (b), When you plot the trajectories be sure to plot the four trajectories that lie along the eigenvectors as well as a few others. The keyboard input option in pplane is useful for this. p. 398, 11, 12, 15. Do not describe the behavior of the solutions in these problems. 1. For the dierential equation in Exercise 10 on p. 282, give the approximating Euler equation. Use it to nd the indicial equation and its roots. Use Theorem 5.6.1 on p. 289 to describe the expected form of the solutions y1 and y2 . Do not nd the coecients! This is meant to be a short exercise.
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2. Repeat Problem 1 for Exercise 3 on p. 282. 3. Do Exercise 10(b) and 10(c) on p. 282.
For Friday, 4/22 Read, p. 401-402 Assignment: p. 409, 1 1. Given that X1 , X2 and X3 are eigenvectors for the following matrix (a) Find the general solution to X = AX . Hint: To nd the eigenvalue, compute AXi . (b) Find the solution X (t) satisfying
1 X (0) = 4 5
2 6 3 5 A = 2 10 6 12 5
1 X1 = 1 2
1 X2 = 2 2
1 X3 = 3 5
2. Find all eigenvalues and eigenvectors for the matrix in Exercise 7 on p. 428 of the text. Explain why the techniques studied so far are insucient to solve this exercise. 3. Given that for the matrix A in Exercise 5 on p. 428, det(A rI ) = (r + 1)(r 2)2 , nd the eigenvalues and eigenvectors. Explain why the techniques studied so far are insucient to solve this exercise.
For Wednesday, 4/27
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Read the note on Exponential Series posted on the web. Do Exercises 1(b),(c) p. 4 of the notes. (These are nilpotent matrices.), p. 14: 7(c)(See Example 1 on p.8 of the notes.), 10(a) (Only nd a fundamental set of solutions. Do not do the part about X (0).) , 13 (The reference to the answers should be to Exercise 12, not Exercise 11.), 14, 15, 18.
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