Differential Equations Solutions to the Review Questions 2021

Solutions to the Review Questions Short Answer/True or False 1. True or False, and explain: (a) If y 0 = y + 2t, then 0 = y + 2t is an equilibrium solution. False. y(t) must be constant. 2 (b) Let dy dt = 1 + y . The Existence and Uniqueness theorem tells us that the solution (for any initial value) will be valid for all t. (If true, say why. If False, solve the DE). False. The E & U Theorem tells that a unique solution will exist for any initial condition, but it does not say on what interval the solution will exist. (c) If y 0 = cos(y), then the solutions are periodic. FALSE. The solution to an autonomous equation can do what? (d) All autonomous equations are separable. True. Any autonomous equation can be written as y 0 = f (y) · 1. (e) All separable equations are exact. True. If the equation is separable, then y 0 = f (y)g(x), which can be written: 1 dy = g(x) f (y) dx ⇒ −g(x) + 1 dy =0 f (y) dx (I’ll let you check that that is exact). 2. The Existence and Uniqueness Theorems: • Linear: y 0 + p(t)y = g(t) at (t0 , y0 ): SOLUTION: See the text. • General Case: y 0 = f (t, y), (t0 , y0 ): SOLUTION: See the text. 3. To solve y 0 = y 1/3 , we separate variables: y −1/3 dy = dt Before going further, it is good practice to note that the previous step is valid, as long as y 6= 0. The case that y = 0 can be taken separately- In fact, we see that y(t) = 0 is an equilibrium solution that satisfies the initial condition. Going on, we integrate: 3 2/3 y = t + C1 2 ⇒ y 2/3 2 = t + C2 3  ⇒ y= 2 t + C2 3 3/2 We can solve for C2 using the initial condition. Of course, the Existence and Uniqueness Theorem cannot be “violated” since it is a theorem, but in this case, the hypotheses are not satisfied: y 0 = f (t, y) ⇒ f (t, y) = y 1/3 In this case, f is continuous at (0, 0) but ∂f /∂y is not. 1 Solve: 1. dy x2 − 2y = dx x SOLUTION: y = C 1 2 x + 2 4 x 2. (x + y) dx − (x − y) dy = 0. Hint: Let v = y/x. Given the hint, rewrite the DE: dy x+y 1 + (y/x) 1+v = = = dx x−y 1 − (y/x) 1−v With the substitution xv = y, we get the substitution for dy/dx: v + xv 0 = y 0 So that the DE becomes: v + xv 0 = 1+v 1−v xv 0 = ⇒ 1+v 1 + v − v(1 − v) 1 + v2 −v = = 1−v 1+v 1−v The equation is now separable: 1−v 1 dv = dx 1 + v2 x Z ⇒ 1 dv − 1 + v2 Z v dv = ln |x| + C 1 + v2 Therefore, tan−1 (v) − 1 ln(1 + v 2 ) = ln |x| + C 2 Lastly, back-substitute v = y/x. 3. dy 2x + y = dx 3 + 3y 2 − x SOLUTION: y(0) = 0. −x2 − xy + 3y + y 3 = 0 4. dy 2xy + y 2 + 1 =− dx x2 + 2xy SOLUTION: x2 y + xy 2 + x = c 5. dy dt = 2 cos(3t) y(0) = 2 SOLUTION: y(t) = 6. y 0 − 21 y = 0 2 3 sin(3t) + 2, and the solution is valid for all time. y(0) = 200. State the interval on which the solution is valid. This is linear and separable. As a linear equation, the solution will be valid on all t (since p(t) = − 12 ). The solution is y(t) = 200e(1/2)t 7. This is separable (or Bernoulli): Z Z y −2 dy = (1 − 2x) dx ⇒ − 1 = x − x2 + C y Put in the initial condition (IC): 6 = 0 + C. Now finish solving explicitly: y(x) = x2 1 1 = −x−6 (x − 3)(x + 2) The solution is valid on the interval (−2, 3). 2 8. y 0 − 21 y = e2t y(0) = 1 SOLUTION: y(t) = 23 e2t + 31 e(1/2)t 9. y 0 = 21 y(3 − y) SOLUTION: y(t) = 3 (1/A)e−(3/2)t + 1 10. sin(2t) dt + cos(3y) dy = 0 SOLUTION: −1 2 cos(2t) + 1 3 sin(3y) = C 11. y 0 = xy 2 SOLUTION: y = 12. 1 −(1/2)x2 − C dy 3 2x − y= (Hint: Let v = y 2 ) dx 2x y SOLUTION: Multiply by y to get 3 2 y = 2x 2x yy 0 − So (following what we did for the general Bernoulli eqn), let v = y 2 , and therefore v 0 = 2yy 0 . Multiply by 2 to get the right form, then substitute 2yy 0 − 3 2 y = 4x x ⇒ v0 − 3 v = 4x x Now it is a standard linear DE. Solving, we get v = −4x3 + Cx3 , and y 2 = −4x3 + Cx3 (We’ll leave in implicit form). 13. yy 00 = (y 0 )2 With the hint, we need to find a substitution for y 00 (and then leave the equation as a DE with unknown function p with variable y): dy = p(y) dt ⇒ d2 y dp dy dp = · = p(y) dt2 dy dt dy Now, substitute in the expressions and divide by yp(y): Z Z dp 1 dp 1 1 = p(y) ⇒ dp = dy y p(y) = (p(y))2 ⇒ dy dy y p y ⇒ ln |p| = ln |y|+C ⇒ p = Ay where A is a constant of integration. Now, substitute again with p = y 0 , and we get: y 0 = Ay ⇒ y = P eAt where P is another constant of integration. We can verify our answer as well: y 0 = AP eAt , and y 00 = A2 P eAt so that yy 00 = P eAt · A2 P eAt = A2 P 2 e2At and this expression is also (y 0 )2 . 14. y 0 + 2y = g(t) with y(0) = 0 and g(t) = 1 on 0 ≤ t ≤ 1 and zero elsewhere. SOLUTION: This is similar to Exercise 33, Section 2.4. In this case, we go ahead and solve starting at time 0: 1 1 y 0 + 2y = 1 ⇒ y(t) = − e−2t 2 2 3 and this is valid for 0 ≤ t ≤ 1. When we hit t = 1, the dynamics change to: y 0 + 2y = 0 ⇒ y(t) = P e−2t Now we will typically choose the constants so that y is continuous. Therefore, using our previous function, y(1) = (1 − e−2 )/2, and our current function: y(1) = P e−2 , or P = e2 − 1 2 Therefore, the overall solution to the DE would be:  (1 − e−2t )/2 y(t) = 2 ((e − 1)/2)e−2t if 0 ≤ t ≤ 1 if 1 > 1 Just for fun, the direction field and solution curve are plotted in Figure 1. Figure 1: Direction field and solution curve for Exercise 14. Note how the solution approaches one equilibrium until g(t) changes, then it goes to the new equilibrium. Misc. dy = 0, and assuming the equation is not currently exact, what differential 1. Given M (x, y) + N (x, y) dx equation would an integrating factor µ(x) solve, if µ depends on x alone? SOLUTION: Let µ(M + N y 0 ) = 0. To be exact, we must have µx M + µMx = µy N + µNy If further, µ(x) depends only on y, then this simplifies to: µ0 M + µMx = 0 + µNy ⇒ ⇒ µ0 = µ · Ny − M x M where the fraction is an expression of x alone, so this would be a separable DE in µ. 2. Construct a linear first order differential equation whose general solution is given by: (a) y(t) = t − 3 + C t2 4 SOLUTION: Construct y 0 . The idea will be to produce a linear DE. Therefore, we need to construct y 0 and compare it to y: y 0 = 1 − 2Ct−3 Add this to some multiple (t’s are allowed) of y to get of the arbitrary constant. In this case, 2 6 6 y 0 + y = (1 − 2Ct−3 ) + 2 − + 2Ct−3 = 3 − t t t or, ty 0 + 2y = 3t − 6 (b) y(t) = 2 sin(3t) + Ce−2t SOLUTION: Same idea as before. Try to get y and y 0 together in such a way as to cancel out the arbitrary C: y 0 = 6 cos(3t) − 2Ce−2t so that: y 0 + 2y = 4 sin(3t) + 6 cos(3t). 3. Suppose we want to construct a population model so that there is a logistic population growth(a) That is, there is an environmental carrying capacity of 100. Construct an appropriate (autonomous) model. SOLUTION: In the (y, y 0 ) plane, we are looking at an upside down parabola that goes through y = 0 and y = 100. One model is therefore y 0 = y(100 − y) (b) Using your model, now assume there is a continuous“harvest” (of k units per time period). How does that effect the model- In particular, is there a critical value of k over which the population will be extinct? If so, find it. SOLUTION: This subtracts k from our population: y 0 = y(100 − y) − k We see that for some value of k, the vertex of the upside parabola is exactly on the y−axis. We can solve this by completing the square- That is, we should be able to write y 0 as a perfect square: −y 2 + 100y + k → −(y 2 − 100y + 502 ) = −(y − 50)2 Therefore, k = 502 = 2500. This is the maximum allowable harvest before the ecosystem collapses. 4. We want to construct a new population model. In this case, we have a “minimum population” constraint in that, if the population every goes below this number (call it a constant R1 ), then there is not enough population to recover- The population goes extinct. There is also a larger number that we called the environmental threshold, R2 . We want to find the simplest model that will have the desired behavior. First, noting that it is autonomous, graph y 0 = f (y), then write down a possible polynomial for f . SOLUTION: Create a cubic polynomial (3 equilibria) with zeros at y = 0, R1 , R2 . Also, consider the arrows along the y−axis indicating increasing or decreasing, and we see that the polynomial will having a leading −ky 3 (with k > 0): y 0 = −ky(y − R1 )(y − R2 ) 5. Suppose we have a tank that contains M gallons of water, in which there is Q0 pounds of salt. Liquid is pouring into the tank at a concentration of r pounds per gallon, and at a rate of γ gallons per minute. The well mixed solution leaves the tank at a rate of γ gallons per minute. Write the initial value problem that describes the amount of salt in the tank at time t, and solve: γ dQ = rγ − Q, dt M Q(0) = Q0 The solution is: Q = rM + (Q0 − rM )e−(γ/M )t 5 6. Referring to the previous problem, if let let the system run infinitely long, how much salt will be in the tank? Does it depend on Q0 ? Does this make sense? Note that the differential equation for Q is autonomous, so we could do a phase plot (line with a negative slope). Or, we can just take the limit as t → ∞ and see that Q → rM . This does not necessarily depend on Q0 ; if Q0 starts at equilibrium, rM , then Q is constant. It does make sense. The incoming concentration of salt is r pounds per gallon, so we would expect the long term concentration to be the same, rM/M = r. 7. Modify problem 5 if: M = 100 gallons, r = 2 and the input rate is 2 gallons per minute, and the output rate is 3 gallons per minute. Solve the initial value problem, if Q0 = 50. 3 dQ =4− Q Q(0) = 50 dt 100 − t This goes from being autonomous to linear. In this case, use an integrating factor, R R 1 e p(t) dt = e3 100−t dt = e−3 ln |100−t| = (100 + t)−3 t > 100 Going back to the DE  Q (100 − t)3  = 4(100 − t)−3 ⇒ Q = 2(100 − t) + C(100 − t)3 Continuing, we get: Q(t) = 2(100 − t) − 150 (100 − t)3 1003 8. Suppose an object with mass of 1 kg is dropped from some initial height. Given that the force due to gravity is 9.8 meters per second squared, and assuming a force due to air resistance of 21 v, find the initial value problem (and solve it) for the velocity at time t. The general model is: mv 0 = mg − kv. In this case, m = 1, g = 9.8 and k = 1/2. Therefore, 1 v 0 = 9.8 − v 2 Which is linear (and autonomous). Since the object is being dropped, the initial velocity is zero. Solve it:   v(t) = 19.6 1 − e−(1/2)t 9. Suppose you borrow $10000.00 at an annual interest rate of 5%. If you assume continuous compounding and continuous payments at a rate of k dollars per month, set up a model for how much you owe at time t in years. Give an equation you would need to solve if you wanted to pay off the loan in 10 years. SOLUTION: First, notice that the units of time are mixed- The interest rate is an annual rate, but k is in dollars per month. We should first decide on what the units of time should be. The answer below will assume that we are working with time in years, so that the annual payments are 12k dollars per year (in a continuous fashion). Therefore, the model is S 0 = rS − 12k, where S will be the amount owing, r is the annual interest rate and k is the rate for the continuous payment (per month). Then using S(0) = S0 , we can write the solution as   12k rt 12k + S0 − e S(t) = r r Substituting in the values for r and S0 , and t = 10, we can solve the equation for k: 1 0 = 240k + (10000 − 240k)e 2 Extra: If we go ahead and solve, we get a monthly payment rate of k ≈ $105.90. 6 10. Show that the IVP xy 0 = y − 1, y(0) = 2 has no solution. (Note: Part of the question is to think about how to show that the IVP has no solution). SOLUTION: We’ll try to solve it, and see what happens. The DE is separable. Notice that the function f (x, y) from the E& U theorem is (y − 1)/x, which is not continuous at x = 0... Continuing as usual: Z 1 dy = y−1 Z 1 dx x ⇒ ln |y − 1| = ln |x| + C y − 1 = Ax ⇒ y = Ax + 1 Exponentiate both sides to get There is no choice of A that will satisfy the initial condition, 2=A·0+1 11. Suppose that a certain population grows at a rate proportional to the square root of the population. Assume that the population is initially 400 (which is 202 ), and that one year later, the population is 625 (which is 252 ). Determine the time in which the population reaches 10000 (which is 1002 ) SOLUTION: If P (t) is the population at time t, then the first part of the statement translates to: dP = kP 1/2 dt where k is the constant of proportionality. This is separable (and autonomous), so: Z Z −1/2 P dP = k dt ⇒ P 1/2 = (k/2)t + C Given the initial population, we can solve for C, given the second piece of info, we can solve for k: P (0) = 202 ⇒ 20 = (k/2)(0) + C ⇒ C = 20 and P (1) = 252 gives: ⇒ 25 = (k/2) + 20 k = 10 Therefore, our model is: P (t) = (5t + 20)2 Solving for the last part, P (t) = 1002 , we have ⇒ 100 = 5t + 20 t = 16 12. Consider the sketch below of F (y), and the differential equation y 0 = F (y). (a) Find and classify the equilibrium. SOLUTION: From the sketch given, y = 0 is asymptotically stable and y = 1 is semistable. (b) Find intervals (in y) on which y(t) is concave up. SOLUTION: Examine the intervals y < 0, 0 < y < 1/3, 1/3 < y < 1 and y > 1 separately. The function y will be concave up when dF/dy and F both have the same sign- This happens when F is either increasing and positive (which happens nowhere) or decreasing and negative: 01 (c) Draw a sketch of y on the direction field, paying particular attention to where y is increasing/decreasing and concave up/down. See the figure below. (d) Find an appropriate polynomial for F (y). SOLUTION: One example is y 0 = −y(y − 1)2 7 13. Consider the DE: y dx + (2x − yey ) dy = 0. Show that the equation is not exact, but becomes exact if you assume there is an integrating factor in terms of y alone (Hint: Find the integrating factor first). SOLUTION: Let µ be the integrating factor, and assume µ is a function of y alone. Then (µM )y = µ0 · M + µ · My = yµ0 + µ And (µN )x = 0 · N + µ · 2 Setting these equal, we have the DE: yµ0 = µ Z ⇒ 1 dµ = µ Z dy y or µ = y. We can verify our answer, since y 2 dx + (2xy − y 2 ey ) dy = 0 should now be exact. 14. Newton’s Law of Cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of the body and the environment (which we assume is some constant). Write down a differential equation which represents this statement, then find the general solution. SOLUTION: If u(t) is the temperature of the body at time t, and T is the (constant) environmental temperature (please define your terms!), then we have: u0 = −k(u − T ) ⇒ u(t) = Ce−kt + T This solution makes sense, since the long term behavior is that the temperature reaches T as t → ∞. 15. Given the direction field below, find a differential equation that is consistent with it. SOLUTION: Draw the corresponding figure in the (y, y 0 ) plane first. There we see that y = 0 is unstable (make it a linear crossing), and y = 2 is stable, y = 4 is unstable. From the figure, y 0 = y(y − 2)(y − 4) will work. 16. Consider the direction field below, and answer the following questions: (a) Is the DE possibly of the form y 0 = f (t)? SOLUTION: No. The isoclines would be vertical (consider, for example, a vertical line at t = −3; the slopes are clearly not equal). 8 (b) Is the DE possible of the form y 0 = f (y)? SOLUTION: No. The isoclines would be horizontal (for example, look at a horizontal line at y = 1Some slopes are zero, others are not). (c) Is there an equilibrium solution? (If so, state it): SOLUTION: Yes- At y = 0. (d) Draw the solution corresponding to y(−1) = 1. SOLUTION: Just draw a curve consistent with the arrows shown. 17. Evaluate the following integrals: SOLUTIONS: You should be able to integrate by parts and use partial fractions fairly quickly at this point. Try checking your answers using Maple! Some notes: The first integral should be done using a table, and the last should simplify a lot. Z Z R x 3 2x x e dx dx e−3 dt/t (x − 1)(2 − x) 9