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All Textbook Solutions for Differential Equations

24E In Exercises 16, find the general solution (in scalar form) of the given second-order equation. 1. d2ydt26dydt7y=0 In Exercises 16, find the general solution (in scalar form) of the given second-order equation. 2. d2ydt2dydt12y=0 In Exercises 16, find the general solution (in scalar form) of the given second-order equation. 3. d2ydt2+6dydt+9y=0 4E 5E 6E In Exercises 712, find the solution of the given initial-value problem. 7. d2ydt2+2dydt3y=0 y(0)=6,y(0)=2 In Exercises 712, find the solution of the given initial-value problem. 8. d2ydt2+4dydt5y=0 y(0)=11,y(0)=7 A 9E In Exercises 712, find the solution of the given initial-value problem. 10. d2ydt2+4dydt+20y=0 y(0)=2,y(0)=8 A In Exercises 712 , find the solution of the given initial-value problem. 11. d2ydt28dydt+16y=0 y(0)=3,y(0)=11 A 12E 13E 14E In Exercises 1320, consider harmonic oscillators with mass $m,$ spring constant $k,$ and damping coefficient b . For the values specified, (a) write the second-order differential equation and the corresponding first-order system; (b) find the eigenvalues and eigenvectors of the linear system; (c) classify the oscillator (as underdamped, overdamped, critically damped, or undamped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t) - and v(t) -graphs of the solution with the given initial condition. 15. m=1,k=5,b=4, with initial conditions y(0)=1,v(0)=0 16E In Exercises 13-20, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified, (a) write the second-order differential equation and the corresponding first-order system; (b) find the eigenvalues and eigenvectors of the linear system; (c) classify the oscillator (as underdamped, overdamped, critically damped, or undamped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t) - and v(t) -graphs of the solution with the given initial condition. 17. m=2,k=1,b=3, with initial conditions y(0)=0,v(0)=3 A In Exercises 1320, consider harmonic oscillators with mass m, spring constant k, and damping coefficient b . For the values specified, (a) write the second-order differential equation and the corresponding first-order system: (b) find the eigenvalues and eigenvectors of the linear system; (c) classify the oscillator (as underdamped, overdamped, critically damped, or undamped) and, when appropriate, give the natural period; (d) sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition; and (e) sketch the y(t) - and v(t) -graphs of the solution with the given initial condition. 18. m=9,k=1,b=6, with initial conditions y(0)=1,v(0)=1 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 1E 2E 3E In Exercises 27 , we consider the one-parameter families of linear systems depending on the parameter a . Each family therefore determines a curve in the trace-determinant plane. For each family, (a) sketch the corresponding curve in the trace-determinant plane; (b) in a brief essay, discuss different types of behaviors exhibited by the system as a increases along the real line (unless otherwise noted); and (c) identify the values of a where the type of the system changes. These are the bifurcation values of a 4. dYdt=(aa10)Y 5E In Exercises 2-7, we consider the one-parameter families of linear systems depending on the parameter a. Each family therefore determines a curve in the trace-determinant plane. For each family, (a) sketch the corresponding curve in the trace-determinant plane; (b) in a brief essay, discuss different types of behaviors exhibited by the system as a increases along the real line (unless otherwise noted); and (c) identify the values of a where the type of the system changes. These are the bifurcation values of a 6. dYdt=(20a 3)Y 7E 8E 9E 10E 11E 12E 13E 14E 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 1RE 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE In Exercises 1-8, find the general solution of the given differential equation. 1. d2ydt2dydt6y=e4t 2E 3E 4E 5E In Exercises 18, find the general solution of the given differential equation. 6. d2ydt2+7dydt+10y=e2t 7E 8E In Exercises 912, find the solution of the given initial-value problem. 9. d2ydt2+6dydt+8y=et y(0)=y(0)=0 In Exercises 912, find the solution of the given initial-value problem. 10. d2ydt2+7dydt+12y=3et y(0)=2,y(0)=1 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 1E 2E 3E 4E 5E 6E 7E In Exercises 110, find the general solution of the given equation. 8. d2ydt2+4dydt+20y=cos5t 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 1E 2E 3E 4E 5E 6E 7E 8E 1RE 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE Consider the three systems (i) dxdt=2x+y (ii) dxdt=2x+y (iii) dxdt=2x+y dydt=y+x2dydt=y+x2dydt=yx2 All three have an equilibrium point at (0,0). Which two systems have phase portraits with the same "local picture" near (0, 0)? Justify your answer. [Hint: Very little computation is required for this exercise, but be sure to give a complete justification. ]2E Consider the system dx dt=2x+y dy dt=y+x2 (a) Find the linearized system for the equilibrium point (0,0) (b) Classify (0,0) (as either a source, sink, center, ). (c) Sketch the phase portrait for the linearized system near (0,0) (d) Repeat parts (a)-(c) for the equilibrium point at (2,4)4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E If a nonlinear system depends on a parameter, then the equilibrium points can change as the parameter varies. In other words, as the parameter changes, a bifurcation can occur. Consider the one-parameter system family of systems dx dt=x2a dy dt=y( x 2 +1) where a is the parameter. (a) Show that the system has no equilibrium points if a0 (b) Show that the system has two equilibrium points if a0. (c) Show that the system has exactly one equilibrium point if a=0 (d) Find the linearization of the equilibrium point for a=0 and compute the eigenvalues of this linear system. Remark: The system changes from having no equilibrium points to having two equilibrium points as the parameter a is increased through a=0. We say that the system has a bifurcation at a=0, and that a=0 is a bifurcation value of the parameter.19E 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 1E 2E 3E 4E 5E 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E For the system dxdt=Ydydt=x3x (a)show that the system is Hamiltonian with Hamiltonian function H(x,y)=y22+x22x44 (b) sketch the level sets of H, and (c) sketch the phase portrait for the system. Include a description of all equilibrium points and any saddle connections.2E

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