Applied Differential Equations – Mth 256Archive – Spring 1996 FilesOct 10, 2000This archive contains the tests from Mth 256 Spring 1996. The original test instructions, headers andformatting have not been preserved.Contents1Test1 12Test2 23 Final Exam 43.1 LAPLACE transformtablesfromtheexam........ 43.2 Theexamproblems.................... 54 Contact Information 71Test1Problem 1. (20 points). A 1000 gallon tank initially contains 600 gallons of brine of concentration1.0 oz salt per gallon. Brine of concentration 2.5 oz salt per gallon flows into the tank at 6 gallons perminute. The well-mixed solution is pumped out at the rate of 4 gallons per minute. Find the concentrationof the brine solution in the tank at the very moment that the tank begins to overflow.Problem 2. (20 points). Use the substitution w = y−2to solve the initial value problemy+ y = y3,y(0) =12.Applied Differential Equations Mth 256Problem 3. (20 points). The differential equation2x2+ ydx +x2y − xdy =0has an integrating factor of the form µ = xk. (A) Find the integrating factor µ. (B) Find the solution of thedifferential equation satisfying the initial condition y(1) = 2.Problem 4. (20 points). Solve the (homogeneous) initial value problemxdydx= y + x sec (y/x) ,y(1) =π4.Problem 5. (20 points). Consider an insulated box with internal temperature T . Assume that theambient (external) temperature A is changing linearly (for a while at least), say A = A0+ A1t where A0and A1are constants, and t is time. According to Netwon’s law of cooling we havedTdt= −k (T −A)where k is a constant depending on the insulation of the box. Find the temperature T (t) in terms of t, A0,A1and k. (Do not neglect the arbitrary constant.)2Test2Problem 6. (10 points). Suppose φ(x) is a particular solution of the ordinary differential equationy+p(x)y+q(x)y = g(x). Suppose that y1(x) and y2(x) are complementary solutions. Suppose moreoverthat y1(a)=1, y1(a)=2, y2(a)=−2, y2(a)=2, φ(a)=3and φ(a)=−1.Find a solution y(x) (in terms of y1(x), y2(x) and φ(x)) of the initial value problemy+ p(x)y+ q(x)y = g(x),y(a)=−1,y(a)=1.Problem 7. (40 points). For each of the following linear homogeneous ordinary differential equa-tions find the general solution (fundamental solution) in real form:(A).d2ydx2−dydx− 6 y =0Bent E. Petersen Page 2 Spring 1996Applied Differential Equations Mth 256(B).d3ydx3−dydx=0(C).d2ydx2+4dydx+13y =0(D).d2ydx2+4dydx+4y =0(E).d2ydx2+16y =0(F).D2+42y =0(G). (D − 2)5y =0(H).D2+2D +52y =0Problem 8. (15 points). Find a particular solution2 y+3y− 5 y =2x3+1.Problem 9. (15 points). Find the form of the particular solution given by the method of undeter-mined coefficients for the ordinary differential equationy+2y+5y =6x + xe−x+2e−xcos(2x)+3xexsin(2x).Do not solve for the coefficients.Problem 10. (20 points). Use the method of variation of parameters to find a particular solution tothe ordinary differential equationd2ydx2+4y = 8 sec(2x).Bent E. Petersen Page 3 Spring 1996Applied Differential Equations Mth 2563 Final Exam3.1 LAPLACE transform tables from the examLaplace Transform Exchange FormulaeIf L{f(t)}(s)=F (s) thenLeatf(t)(s)=F(s − a)L{tnf(t)}(s)=(−1)nF(n)(s)Lf(t)t(s)=Z∞sF (r) driff(t)tintegrable at 0LZt0f(r) dr(s)=F (s)sLdfdt(s)=sF(s) − f (0)if f cont. on [0, ∞)Ld2fdt2(s)=s2F (s) − sf(0) − f(0)if f, fcont. on [0 , ∞)L{u(t − a)f(t − a)}(s)=e−asF (s)u = unit stepL{f(at)}(s)=1aF (sa).If L{f(t)}(s)=F (s) and L{g(t)}(s)=G(s) then L{(f ∗ g)(t)}(s)=F (s)G(s).Bent E. Petersen Page 4 Spring 1996Applied Differential Equations Mth 256Sample Laplace TransformsLeat(s)=1s − aL{tn}(s)=n!sn+1L{cos ωt}(s)=ss2+ ω2L{sin ωt}(s)=ωs2+ ω2Leatcos ωt(s)=s − a(s − a)2+ ω2Leatsin ωt(s)=ω(s − a)2+ ω2Ln√to(s)=√π2s3/2Ltneat(s)=n!(s − a)n+1.3.2 The exam problemsProblem 11. (25 points). A 100 gallon tank contains 100 gallons of brine of concentration 1.0 ozsalt per gallon. Brine of concentration 2.0 oz salt per gallon flows into the tank at 2 gallons per minute. Thewell-mixed solution is pumped out at the same rate. Find the concentration of the brine solution in the tankat the end of 30 minutes.Problem 12. (25 points). Solve the initial value problemdpdt=e−2tp (1 − p),p(0) = 1/2.Bent E. Petersen Page 5 Spring 1996Applied Differential Equations Mth 256Problem 13. (20 points). (A) Solve the exact ordinary differential equation2xy − sec2(x)dx +x2+2y +cos(y)dy =0.(B) The differential equation(2y − 6x) dx +3x − 4x2y−1dy =0has an integrating factor of the form µ = xpyqwhere p and q are integers. Find p and q.Problem 14. (20 points). Solve the (homogeneous) ordinary differential equationx2dydx= xy + y2+ x2.Problem 15. (20 points). For each of the following linear homogeneous ordinary differential equa-tions find the general solution (fundamental solution) in real form:(A).d2ydx2− 2dydx+5y =0(B).d4ydx4+d2ydx2=0(C).d2ydx2− 2dydx+ y =0(D).d2ydx2+5dydx+6y =0Problem 16. (20 points). Find the form of the particular solution given by the method of undeter-mined coefficients for the ordinary differential equationy+2y− 3 y = x3+ x + x2ex+excos(x)+x2e−3x+e−3xsin(x).Do not solve for the coefficients.Bent E. Petersen Page 6 Spring 1996Applied Differential Equations Mth 256Problem 17. (20 points). Use the method of variation of parameters to find a particular solution tothe ordinary differential equationd2ydx2+ y =sec3(x).Problem 18. (25 points). If y(t) is the solution to the initial value problem3d2ydt2− 5dydt+13y =cos(2t),y(0) = 3,y(0) = −2,find the Laplace transform Y (s)=L{y(t)} of y. Note it is not necessary to solve the initial value problem.Problem 19. (25 points). Find the inverse Laplace transformL−16 s2− s − 17(s +2)(s2− 1).4 Contact InformationThe contact information below is accurate as of Oct 10, 2000.Copyrightc 2000 by Bent E. Petersen. Permission is granted to duplicate thisdocument for non–profit educational purposes provided that no alterations aremade and provided that this copyright notice is preserved on all copies.Bent E. Petersenphone numbersDepartment of Mathematicsoffice (541) 737-5163Oregon State Universityhome (541) 753-1829Corvallis, OR 97331-4605fax (541)