Solution Found!
Consider the system in Exercise 4. (a) Find the general
Chapter , Problem 5(choose chapter or problem)
Consider the system in Exercise 4. (a) Find the general solution of the equation dx/dt = x. [Hint: This is as easy as it looks.] (b) Using the solution to part (a) in place of x, find the general solution of the equation dy dt = 4x3 + y. [Hint: Note that this system is partially decoupled, so this part of the exercise is really asking you to follow the techniques described in Sections 1.8 and 2.4.] (c) Use the results from parts (a) and (b) to form the general solution of the system. (d) Find the solution curves of the system that tend toward the origin as t . (e) Find the solution curves of the system that tend toward the origin as t . (f) Sketch the solution curves in the phase plane corresponding to these solutions. These are the separatrices. (g) Compare the sketch of the linearized system that you obtained in Exercise 4 with a sketch of the separatrix solutions for the equilibrium point at the origin for this system. In what ways are the two pictures the same? How do they differ?
Questions & Answers
QUESTION:
Consider the system in Exercise 4. (a) Find the general solution of the equation dx/dt = x. [Hint: This is as easy as it looks.] (b) Using the solution to part (a) in place of x, find the general solution of the equation dy dt = 4x3 + y. [Hint: Note that this system is partially decoupled, so this part of the exercise is really asking you to follow the techniques described in Sections 1.8 and 2.4.] (c) Use the results from parts (a) and (b) to form the general solution of the system. (d) Find the solution curves of the system that tend toward the origin as t . (e) Find the solution curves of the system that tend toward the origin as t . (f) Sketch the solution curves in the phase plane corresponding to these solutions. These are the separatrices. (g) Compare the sketch of the linearized system that you obtained in Exercise 4 with a sketch of the separatrix solutions for the equilibrium point at the origin for this system. In what ways are the two pictures the same? How do they differ?
ANSWER:Step 1 of 7
(a) The general solution of the equation
is obtained as follows.
On integrating both sides yields us
Let us take an initial condition . Then we have . Thus the general solution of the equation is