Math 244 Review Solutions Exam 2

Review Solutions Exam 2 1. State the Existence and Uniqueness theorem for linear, second order differential equations (non-homogeneous is the most general form): SOLUTION: Let y 00 + p(t)y 0 + q(t)y = g(t), with y(t0 ) = y0 and y 0 (t0 ) = v0 . Then if p, q and g are all continuous on an open interval I containing t0 , a unique solution exists to the IVP, valid for all t in I. 2. True or False? (a) The characteristic equation for y 00 + y 0 + y = 1 is r2 + r + 1 = 1 SOLUTION: False. The characteristic equation is for the homogeneous equation, r2 + r + 1 = 0 (b) The characteristic equation for y 00 + xy 0 + ex y = 0 is r2 + xr + ex = 0 SOLUTION: False. The characteristic equation was defined only for DEs with constant coefficients. (c) The function y = 0 is always a solution to a second order linear homogeneous differential equation. SOLUTION: True. It is true generally- If L is a linear operator, then L(0) = 0. (d) In using the Method of Undetermined Coefficients, the ansatz yp = (Ax2 + Bx + C)(D sin(x) + E cos(x)) is equivalent to yp = (Ax2 + Bx + C) sin(x) + (Dx2 + Ex + F ) cos(x) SOLUTION: False- We have to be able to choose the coefficients for each polynomial (for the sine and cosine) independently of each other. In the form: (Ax2 + Bx + C)(D sin(x) + E cos(x)) the polynomials for the sine and cosine are constant multiples of each other, which may not necessarily hold true. That’s why we need one polynomial for the sine, and one for the cosine (so the second guess is the one to use). (e) Consider the function: y(t) = cos(t) − sin(t) Then amplitude is 1, the period is 1 and the phase shift is 0. SOLUTION: False. For this question to make sense, we have to first write the function as R cos(ω(t − δ)). In this case, the amplitude is R: q √ R = 12 + (−1)2 = 2 The period is 2π (the circular frequency, or natural frequency, is 1), and the phase shift δ is: π tan(δ) = −1 ⇒ δ = − 4 1 3. Find values of a for which any solution to: y 00 + 10y 0 + ay = 0 will tend to zero (that is, limt→∞ y(t) = 0. SOLUTION: Use the characteristic equation and check the 3 cases (for the discriminant). That is, √ −10 ± 100 − 4a r2 + 10r + a = 0 ⇒ r = 2 We check some special cases: • If 100 − 4a = 0 (or a = 25), we get a double root, r = −5, −5, or yh = e−5t (C1 + C2 t), and all solutions tend to zero. • If the roots are complex, then we can write r = −5 ± βi, and we get yh = e−5t (C1 cos(βt) + C2 sin(βt)) and again, this will tend to zero for any choice of C1 , C2 . • In the case that a < 25, we have to be a bit careful. While it is true that both roots will be real, we also want them to both be negative for all solutions to tend to zero. √ – When will they both be negative? If 100 − 4a < 100 (or 100 − 4a < 10). This happens as long as a > 0. – If a = 0, the roots will be r = −10, 0, and yh = C1 e−10t + C2 - Therefore, I could choose C1 = 0 and C2 6= 0, and my solution will not go to zero. – If a < 0, the roots will be mixed in sign (one positive, one negative), so the solutions will not all tend to zero. CONCLUSION: If a > 0, all solutions to the homogeneous will tend to zero. 4. • Compute the Wronskian between f (x) = cos(x) and g(x) = 1. SOLUTION: W (cos(x), 1) = sin(x) • Can these be two solutions to a second order linear homogeneous differential equation? Be specific. (Hint: Abel’s Theorem) SOLUTION: Abel’s Theorem tells us that the Wronskian between two solutions to a second order linear homogeneous DE will either be identically zero or never zero on the interval on which the solution(s) are defined. Therefore, as long as the interval for the solutions do not contain a multiple of π (for example, (0, π), (π, 2π), etc), then it is possible for the Wronskian for two solutions to be sin(x). 2 5. Construct the operator associated with the differential equation: y 0 = y 2 − 4. Is the operator linear? Show that your answer is true by using the definition of a linear operator. SOLUTION: The operator is found by getting all terms in y to one side of the equation, everything else on the other. In this case, we have: L(y) = y 0 − y 2 This is not a linear operator. We can check using the definition: L(cy) = cy 0 − c2 y 2 6= cL(y) Furthermore, L(y1 + y2 ) = (y10 + y20 ) − (y1 + y2 )2 6= L(y1 ) + L(y2 ) 6. Find the solution to the initial value problem:    3t u + u = 3(2π − t)   0 00 if 0 ≤ t ≤ π if π < t < 2π if t ≥ 2π u(0) = 0 u0 (0) = 0 SOLUTION: Without regards to the initial conditions, we can solve the three nonhomogeneous equations. In each case, the homogeneous part of the solution is c1 cos(t)+ c2 sin(t). • u00 + u = 3t. We would start with yp = At + B. Substituting, we get: At + B = 3t, so A = 3 and B = 0- Therefore, the general solution in this case is: u(t) = c1 cos(t) + c2 sin(t) + 3t • u00 + u = 6π − 3t. From our previous analysis, the solution is: u(t) = c1 cos(t) + c2 sin(t) + 6π − 3t • The last part is just the homogeneous equation. The only thing left is to find c1 , c2 in each of the three cases so that the overall function u is continuous: • u(0) = 0, u0 (0) = 0 ⇒ u(t) = −3 sin(t) + 3t 0≤t≤π • u(π) = 3π and u0 (π) = 6, so: u(t) = −9 sin(t) + 6π − 3t 3 π < t < 2π • u(2π) = 0, u0 (2π) = −12: u(t) = −12 sin(t) 7. Solve: u00 + ω02 u = F0 cos(ωt), Undetermined Coefficients. t ≥ 2π u(0) = 0 u0 (0) = 0 if ω 6= ω0 using the Method of SOLUTION: The characteristic equation is: r2 + ω02 = 0, or r = ±ω0 i. Therefore, uh = C1 cos(ω0 t) + C2 sin(ω0 t) Using the Method of Undetermined Coefficients, up = A cos(ωt) + B sin(ωt), and we put that into the DE: +Bω02 sin(ωt) ω02 u = Aω02 cos(ωt) 00 2 −Bω 2 sin(ωt) u = −Aω cos(ωt) F0 cos(ωt) = A(ω02 − ω 2 ) cos(ωt) +B(ω02 − ω 2 ) sin(ωt) Therefore, A= ω02 F0 − ω2 B=0 so that the overall solution is: u(t) = C1 cos(ω0 t) + C2 sin(ω0 t) + F0 cos(ωt) ω02 − ω 2 0 Put in the initial conditions u(0) = 0 and u0 (0) = 0 to see that C1 = − ω2F−ω 2 and 0 C2 = 0. 8. Compute the solution to: u00 + ω02 u = F0 cos(ω0 t) u(0) = 0 u0 (0) = 0 two ways: • Start over, with Method of Undetermined Coefficients SOLUTION: The part that changes is the particular part of the solution- We have to multiply by t: Let up = At cos(ω0 t) + Bt sin(ω0 t). Then: ω02 up = (Aω02 t ) cos(ω0 t) +(Bω02 t ) sin(ω0 t) 00 2 up = (−Aω0 t + 2Bω0 ) cos(ω0 t) +(−Bω02 t − 2Aw) sin(ω0 t) F0 cos(ω0 t) = 2Bω0 cos(ω0 t) −2Aω0 sin(ω0 t) Therefore, B = F0 2ω0 and A = 0, so that u(t) = C1 cos(ω0 t) + C2 sin(ω0 t) + F0 t sin(ω0 t) 2ω0 Taking into account the initial conditions, we get C1 = C2 = 0. 4 • Take the limit of your answer from Question 6 as ω → ω0 (Typo: Should be Question 7). SOLUTION: F0 (cos(ωt) − cos(ω0 t) lim =? ω→ω0 ω02 − ω 2 We can use l’Hospital’s Rule (differentiate with respect to ω!): = lim ω→ω0 −F0 t sin(ωt) F0 = sin(ω0 t) −2ω 2ω0 t 9. For the following question, recall that the acceleration due to gravity is 32 ft/sec2 . An 8 pound weight is attached to a spring from the ceiling. When the weight comes to rest at equilibrium, the spring has been stretched 2 feet. The damping constant for the system is 1−lb-sec/ft. If the weight is raised 6 inches above equilibrium and given an upward velocity of 1 ft/sec, find the equation of motion for the weight. SOLUTION: First the constants. Since mg − kL = 0, we find that mg = 8, so 2k = 8, or k = 4. We are given that γ = 1, and since 8 = mg, then m = 8/32 = 1/4. 1 00 y + y 0 + 4y = 0 4 y(0) = − 1 2 y 0 (0) = −1 Or we could write: y 00 + 4y 0 + 16y = 0. Solving the characteristic equation, we get √ √ √ −4 ± 4 3 i −4 ± 42 − 4 · 16 2 = = −2 ± 2 3 i r + 4r + 16 = 0 ⇒ r = 2 2 Therefore, the general solution is:  √  √ y(t) = e−2t C1 cos(2 3t) + C2 sin(2 3 t Solving the IVP, differentiate to get the equations for C1 , C2 :   √ √  √ √ √  y 0 = −2e−2t C1 cos(2 3t) + C2 sin(2 3 t + 2 3e−2t C2 cos(2 3t) − C1 sin(2 3 t Therefore, − 12 = C1 √ −1 = −2C1 + 2 3C2 10. Given that y1 = 1 t ⇒ C1 = − solves the differential equation: t2 y 00 − 2y = 0 Find a fundamental set of solutions. SOLUTION: I like using the Wronskian for these5 1 2 1 C2 = − √ 3 First, rewrite the differential equation in standard form: y 00 − 2 y=0 t2 Then p(t) = 0 and W (y1 , y2 ) = Ce0 = C. On the other hand, the Wronskian is: 1 1 W (y1 , y2 ) = y20 + 2 y2 t t Put these together: 1 0 1 y2 + 2 y2 = C t t 1 y20 + y2 = Ct t The integrating factor is t, (ty2 )0 = Ct2 ty2 = C1 t3 + C2 ⇒ ⇒ C1 t2 + Notice that we have both parts of the homogeneous solution, y1 = C2 t 1 t and y2 = t2 . Alternative Solution: Use Reduction of Order, where we assume that y2 = vy1 = v t ⇒ y20 = v0t − v v0 v = − 2 2 t t t ⇒ y200 = v 00 v0 v −2 2 +2 3 t t t ⇒ y2 = C3 t2 + Substituting this back into the DE, we get tv 00 − 2v 0 = 0 ⇒ v0 = C1 t2 ⇒ v= C1 3 t + C2 3 C2 t and again we see that we can take y2 = t2 . 11. Suppose a mass of 0.01 kg is suspended from a spring, and the damping factor is γ = 0.05. If there is no external forcing, then what would the spring constant have to be in order for the system to critically damped? underdamped? SOLUTION: We can find the differential equation: 0.01u00 + 0.05u0 + ku = 0 ⇒ u00 + 5u0 + 100ku = 0 Then the system is critically damped if the discriminant (from the quadratic formula) is zero: 25 1 b2 − 4ac = 25 − 4 · 100k = 0 ⇒ k = = 400 16 The system is underdamped if the discriminant is negative: 25 − 400k < 0 ⇒ k> 1 16 12. Give the full solution, using any method(s). If there is an initial condition, solve the initial value problem. 6 (a) y 00 + 4y 0 + 4y = t−2 e−2t Using the Variation of Parameters, yp = u1 y1 + u2 y2 , we have: y1 = e−2t y2 = te−2t g(t) = e−2t t2 with a Wronskian of e−4t . You should find that: u01 = − 1 t u02 = 1 t2 1 t −2t −2t so yp = − ln(t)e − e . This last term is part of the homogeneous solution, so this simplifies to − ln(t)e−2t . Now that we have all the parts, u1 = − ln(t) u2 = − y(t) = e−2t (C1 + C2 t) − ln(t)e−2t You should note that here we have to use Variation of Parameters, since the forcing function is not one of the forms for Method of Undetermined Coefficients. (b) y 00 − 2y 0 + y = tet + 4, y(0) = 1, y 0 (0) = 1. With the Method of Undetermined Coefficients, we first get the homogeneous part of the solution, yh (t) = et (C1 + C2 t) Now we construct our ansatz (Multiplied by t after comparing to yh ): g1 = tet ⇒ yp1 = (At + B)et · t2 Substitute this into the differential equation to solve for A, B: yp0 1 = (At3 + (3A + B)t2 + 2Bt)et yp1 = (At3 + Bt2 )et yp001 = (At3 + (6A + B)t2 + (6A + 4B)t + 2B)et Forming yp001 − 2yp0 1 + yp1 = tet , we should see that A = yp1 = 61 t3 et . The next one is a lot easier! yp2 = A, so A = 4, and: 1 6 and B = 0, so that 1 y(t) = et (C1 + C2 t) + t3 et + 4 6 with y(0) = 1, C1 = −3. Solving for C2 by differentiating should give C2 = 4. The full solution:   1 3 y(t) = et t + 4t − 3 + 4 6 7 (c) y 00 + 4y = 3 sin(2t), y(0) = 2, y 0 (0) = −1. The homogeneous solution is C1 cos(2t) + C2 sin(2t). Just for fun, you could try Variation of Parameters. We’ll outline the Method of Undetermined Coefficients: yp = (A sin(2t) + B cos(2t))t = At sin(2t) + Bt cos(2t) yp00 = (−4At − 4B) sin(2t) + (4A − 4Bt) cos(2t) taking yp00 + 4yp = 3 sin(2t), we see that A = 0, B = − 34 , so the solution is: 3 y = c1 cos(2t) + c2 sin(2t) − t cos(2t) 4 With y(0) = 2, c1 = 2. Differentiating to solve for c2 , we find that c2 = −1/8. (d) y 00 + 9y = N X bm cos(mπt) m=1 The homogeneous part of the solution is C1 cos(3t) + C2 sin(3t). We see that 3 6= mπ for m = 1, 2, 3, . . .. The forcing function is a sum of N functions, the mth function is: gm (t) = bm cos(mπt) ⇒ ypm = A cos(mπt) + B sin(mπt) Differentiating, yp00m = −m2 π 2 A cos(mπt) − m2 π 2 B sin(mπt) so that yp00m + 9ypm = (9 − m2 π 2 )A cos(mπt) + (9 − m2 π 2 )B sin(mπt). Solving for the coefficients, we see that A = bm /(9 − m2 π 2 ) and B = 0. Therefore, the full solution is: y(t) = C1 cos(3t) + C2 sin(3t) + N X bm cos(mπt) 2 2 m=1 9 − m π 13. Rewrite the expression in the form a + ib: (i) 2i−1 (ii) e(3−2i)t (iii) eiπ NOTE for the SOLUTION: Remember that for any non-negative number A, we can write A = eln(A) . • 2i−1 = eln(2 i−1 ) = e(i−1) ln(2) = e− ln(2) ei ln(2) = 12 (cos(ln(2)) + i sin(ln(2))) • e(3−2i)t = e3t e−2ti = e3t (cos(−2t) + i sin(−2t)) = e3t (cos(2t) − i sin(2t)) (Recall that cosine is an even function, sine is an odd function). • eiπ = cos(π) + i sin(π) = −1 14. Write a + ib in polar form: (i) −1 − √ 3i (ii) 3i (iii) −4 (iv) SOLUTIONS: 8 √ 3−i (i) r = √ 1 + 3 = 2, θ = −2π/3 (look at its graph, use 30-60-90 triangle): √ 2π −1 − 3i = 2e− 3 i π (ii) 3i = 3e 2 i (iii) −4 = 4eπi √ π (iv) 3 − i = 2e− 6 i 15. Find a second order linear differential equation with constant coefficients whose general solution is given by: 1 y(t) = C1 + C2 e−t + t2 − t 2 SOLUTION: Work backwards from the characteristic equation to build the homogeneous DE (then figure out the rest): The roots to the characteristic equation are r = 0 and r = −1. The characteristic equation must be r(r + 1) = 0 (or a constant multiple of that). Therefore, the differential equation is: y 00 + y 0 = 0 For yp = 21 t2 − t to be the particular solution, yp00 + yp0 = (1) + (t − 1) = t so the full differential equation must be: y 00 + y 0 = t 16. Determine the longest interval for which the IVP is certain to have a unique solution (Do not solve the IVP): t(t − 4)y 00 + 3ty 0 + 4y = 2 y(3) = 0 y 0 (3) = −1 SOLUTION: Write in standard form firsty 00 + 4 2 3 0 y + y= t−4 t(t − 4) t(t − 4) The coefficient functions are all continuous on each of three intervals: (−∞, 0), (0, 4) and (4, ∞) Since the initial time is 3, we choose the middle interval, (0, 4). 9 17. Let L(y) = ay 00 + by 0 + cy for some value(s) of a, b, c. If L(3e2t ) = −9e2t and L(t2 + 3t) = 5t2 + 3t − 16, what is the particular solution to: L(y) = −10t2 − 6t + 32 + e2t SOLUTION: This purpose of this question is to see if we can use the properties of linearity to get at the answer. We see that: L(3e2t ) = −9e2t , or L(e2t ) = −3e2t so: 1 L − e2t = e2t 3   And for the second part, L(t2 + 3t) = 5t2 + 3t − 16 L((−2)(t2 + 3t)) = −10t2 + 6t − 32 ⇒ The particular solution is therefore: 1 yp (t) = −2(t2 + 3t) − e2t 3 since we have shown that 1 L −2(t2 + 3t) − e2t = −10t2 + 6t − 32 + e2t 3   18. Use Variation of Parameters to find a particular solution to the following, then verify your answer using the Method of Undetermined Coefficients: 4y 00 − 4y 0 + y = 16et/2 SOLUTION: For the Variation of Parameters, write in standard form first, then compute y1 , y2 , W (y1 , y2 ), and then use the formulas (they will be given to you): 1 y 00 − y 0 + y = 4et/2 4 ⇒ r2 − r + 1 =0 4 ⇒ r = 1/2, 1/2 Therefore, y1 = et/2 y2 = tet/2 W (y1 , y2 ) = et And, 4et −4tet 0 = −4t u = 2 et et so that u1 = −2t2 and u2 = 4t. Putting these back into our ansatz, u01 = yp = −2t2 et/2 + 4t(t2 et/2 ) = 2t2 et/2 You can verify this solution using the Method of Undetermined Coefficients. 10 19. Compute the Wronskian of two solutions of the given DE without solving it: x2 y 00 + xy 0 + (x2 − α2 )y = 0 Using Abel’s Theorem (and writing the DE in standard form first): 1 x2 − α 2 y 00 + y 0 + y=0 x x2 Therefore, − W (y1 , y2 ) = Ce R 1 x dx = C x 20. If y 00 − y 0 − 6y = 0, with y(0) = 1 and y 0 (0) = α, determine the value(s) of α so that the solution tends to zero as t → ∞. SOLUTION: Solving as usual gives:  y(t) = α + 2 3t 3 − α −2t e + e 5 5    so to make sure the solutions tend to zero, α = −2 (to zero out the second term). 21. Give the general solution to y 00 + y = 1 sin(t) +t SOLUTION: If we can use any method, I would use Method of Undetermined Coefficients for g2 (t) = t and Variation of Parameters for g1 (t) = 1/ sin(t). Let’s do the Variation of Parameters first: y1 = cos(t) y2 = sin(t) W (y1 , y2 ) = 1 g(t) = 1 sin(t) Therefore, u01 = −1 u02 = cos(t) sin(t) Continuing, u1 = −t u2 = ln(sin(t)) and yp = −t cos(t) + ln(sin(t)) sin(t) For the other part, we take yp = At + B, and find that yp = t, so that y(t) = C1 cos(t) + C2 sin(t) + t − t cos(t) + ln(sin(t)) sin(t) 22. A mass of 0.5 kg stretches a spring an additional 0.05 meters past its natural length. (i) Find the spring constant. (ii) Does a stiff spring have a large spring constant or a small spring constant (explain). 11 SOLUTION: We use Hooke’s Law at equilibrium: mg − kL = 0, or k= mg 4.9 = = 98 L 0.05 For the second part, a stiff spring will not stretch, so L will be small (and k would therefore be large), and a spring that is not stiff will stretch a great deal (so that k will be smaller). 23. A mass of 12 kg is attached to a spring with spring constant 2 (kg/sec2 ). The spring is pulled down an additional 1 meter then released. Find the equation of motion if the damping constant is c = 2 as well: SOLUTION: Just substitute in the values 1 00 u + 2u0 + 2u = 0 2 Pulling down the spring and releasing: u(0) = 1, u0 (0) = 0 (Down is positive) 24. Write the following as R cos(ωt − δ): 2 cos(3t) + sin(3t). √ √ SOLUTION: R = 22 + 12 = 5 and δ = tan−1 (1/2). so we have √ 5 cos(3t − tan−1 (1/2)) 25. Find the general solution by complexification: y 00 + 3y 0 + 2y = cos(t). SOLUTION: Note that the question asks for the general solution, so we do need both the homogeneous and particular parts of the solution. For the homogeneous part, r2 + 3r + 2 = 0 ⇒ (r + 1)(r + 2) = 0 ⇒ yh = C1 e−t + C2 e−2t For the particular part, cos(t) by eit and make the ansatz yp = Aeit Aeit (−1 + 3i + 2) = eit ⇒ A= 1 1 + 3i You can multiply Aeit out and find the real part, or we can write the particular part of the solution as R cos(t − δ), which is much easier. R= √ 1 1 =√ 2 +3 10 δ = tan−1 (3/1) 12 Therefore the full general solution is given by: 1 y(t) = C1 e−t + C2 e−2t + √ cos(t − tan−1 (3)) 10 Alternate Solution: If you multiplied Aeit out, then took the real part, you would get: y(t) = C1 e−t + C2 e−2t + 12 1 3 cos(t) + sin(t) 10 10 26. Find the amplitude and phase angle only for the particular solution to u00 + u0 + 2u = cos(t). SOLUTION: We make yp = Aeit , complexify the right hand side of the equation, and find the constant A first. Substituting into the differential equation, we have Aeit (−1 + i + 2) = eit ⇒ A= 1 1+i Therefore, R= √ 1 1 √ = 12 + 12 2 δ = tan−1 (1) = π 4 27. Consider u00 + u0 + u = cos(ωt). Find the value of ω that will maximize the amplitude of the response. SOLUTION: This is almost the same as the last problem, except yp = Aeiωt . Substituting that into our equation gives us: Aeiωt (−ω 2 + iω + 1) = eiωt ⇒ A= 1 (1 − ω 2 ) + iω Now, the amplitude of the response function is given by R= q 1 (1 − ω 2 )2 + ω 2 To maximize R, we differentiate it with respect to ω, and set it equal to zero. Typically, we have to ensure that we have a max and not a min, but I’ll allow you to skip that here. Remember our “shortcut” that said we only have to differentiate the function under the radical sign and set that equal to zero. R= 1 q ⇒ f (ω) f 0 (ω) R =− =0 2(f (ω))3/2 0 ⇒ f 0 (ω) = 0 Therefore, f (ω) = (1 − ω 2 )2 + ω 2 √ from which we get ω = 1/ 2. f 0 (ω) = 2(1 − ω 2 )(−2ω) + 2ω = 0 ⇒ 28. Match the solution curve to its IVP (There is one DE with no graph, and one graph with no DE- You should not try to completely solve each DE). (a) 5y 00 + y 0 + 5y = 0, y(0) = 10, y 0 (0) = 0 (Complex roots, solutions go to zero) Graph C (b) y 00 + 5y 0 + y = 0, y(0) = 10, y 0 (0) = 0 (Exponentials, solutions go to zero) Graph D 13 (c) y 00 + y 0 + 54 y = 0, y(0) = 10, y 0 (0) = 0 NOT USED (d) 5y 00 + 5y = 4 cos(t), y(0) = 0, y 0 (0) = 0 (Pure Harmonic) Graph B (e) y 00 + 21 y 0 + 2y = 10, y(0) = 0, y 0 (0) = 0 (Complex roots to homogeneous solution, constant particular solution) Graph E SOLUTION: If the graphs are labeled: Top row: A, B, second row: C, D, and last row E, then the graphs are given above. 14