18. Convert the second-order equationd²y0dt2into a first-order system using v =dy/dt as usual.(a) Find the general solution for the dv/dt equation.(b) Substitute this solution into the dy/dt equation, and find the general solutionof the system.(c) Sketch the phase portrait of the system.

In this problem, we need to solve a second order differential equation by using a given variable…

Answered: 18. Convert the second-order equation…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3t Verify that the pair x(t) = ³t, y(t) = ¹ is a solution to the given system below. Sketch the trajectory of the given solution in the phase plane. e dx dt Ay 3 dy dt Verify that the pair x(t), y(t) is a solution of the system. Select the correct choice below and fill in any answer boxes to complete your choice. (Type expressions using t as the variable.) O A. 3t The pair x(t) = ³t, y(t) = ¹ is a solution because in terms of t. x(t) dt = | 10- =3y³. and i Sy(t) dt = | and y' (t) = OB. The pair x(t) = e³t, y(t)=e¹ is a solution because in terms of t, x'(t)= 3t Sketch the trajectory of the given solution in the phase plane. Choose the correct graph below. O A. 0- =y 100 O D. A) 10+ H 0 10 Q Q Q Q ✔ In terms of x and y, x'(t) = O B. Ay 100- O E. Ay In terms of x and y, x(t) dt = and y(t) dt = | || and y'(t)= 100+ 10 10 Q Q Ⓡ O C. A 10- 0 O F. Ay 10+ 0- 10 10 Q Q ✔ Q
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Consider the following system of equations. dx dt dy =y y=x-x²-y dt (a) Find the equilibrium points of the system. smallest x-value (x, y) = largest x-value (x, y) = (x, y) = 5 (b) Using technology, as directed by your instructor, sketch the direction field and phase portrait of the system. (c) Briefly describe the behavior of typical solutions, As t increases, typical solutions spiral in the Select direction Select one of the outside two solutions depending on the initial condition.
2.5 Consider the system x' = (² 9) x. Find the special values of a for which the phase plane changes type. (Assume det A=0).
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EXERSICE 1 Find the 2nd order linear differential equation whose fundamental set of solutions is {y₁ (x) = cosbx, y₂ (x) = sinbx}, bЄR. (Hint: use the definition and properties of the Wronski determinant)
Solve (a) please
3 please
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18. Convert the second-order equation d²y 0 dt2 into a first-order system using v = dy/dt as usual. (a) Find the general solution for the dv/dt equation. (b) Substitute this solution into the dy/dt equation, and find the general solution of the system. (c) Sketch the phase portrait of the system.