# 4.4.1: Autonomous Second Order Equations (Exercises) - Mathematics

## Q4.4.1

In Exercises 4.4.1-4.4.4 find the equations of the trajectories of the given undamped equation. Identify the equilibrium solutions, determine whether they are stable or unstable, and plot some trajectories.

1. (y''+y^3=0)

2. (y''+y^2=0)

3. (y''+y|y|=0)

4. (y''+ye^{-y}=0)

## Q4.4.2

In Exercises 4.4.5–4.4.8 find the equations of the trajectories of the given undamped equation. Identify the equilibrium solutions, determine whether they are stable or unstable, and find the equations of the separatrices (that is, the curves through the unstable equilibria). Plot the separatrices and some trajectories in each of the regions of Poincaré plane determined by them.

5. (y''-y^3+4y=0)

6. (y''+y^3-4y=0)

7. (y''+y(y^2-1)(y^2-4)=0)

8. (y''+y(y-2)(y-1)(y+2)=0)

## Q4.4.3

In Exercises 4.4.9–4.4.12 plot some trajectories of the given equation for various values (positive, negative, zero) of the parameter a. Find the equilibria of the equation and classify them as stable or unstable. Explain why the phase plane plots corresponding to positive and negative values of a differ so markedly. Can you think of a reason why zero deserves to be called the critical value of (a)?

9. (y''+y^2-a=0)

10. (y''+y^3-ay=0)

11. (y''-y^3+ay=0)

12. (y''+y-ay^3=0)

## Q4.4.4

In Exercises 4.4.13-4.4.18 plot trajectories of the given equation for (c = 0) and small nonzero (positive and negative) values of (c) to observe the effects of damping.

13. (y''+cy'+y^3=0)

14. (y''+cy'-y=0)

15. (y''+cy'+y^3=0)

16. (y''+cy'+y^2=0)

17. (y''+cy'+y|y|=0)

18. (y''+y(y-1)+cy=0)

## Q4.4.5

19. The van der Pol equation

[y''-mu(1-y^2)y'+y=0, ag{A}]

where (mu) is a positive constant and (y) is electrical current (Section 6.3), arises in the study of an electrical circuit whose resistive properties depend upon the current. The damping term (-mu(1-y^2)y') works to reduce (|y|) if (|y|<1) or to increase (|y|) if (|y|>1). It can be shown that van der Pol’s equation has exactly one closed trajectory, which is called a limit cycle. Trajectories inside the limit cycle spiral outward to it, while trajectories outside the limit cycle spiral inward to it (Figure [figure:4.4.16}). Use your favorite differential equations software to verify this for (mu=.5,1.1.5,2). Use a grid with (-4

20. Rayleigh’s equation,

[y''-mu(1-(y')^2/3)y'+y=0]

also has a limit cycle. Follow the directions of Exercise 4.4.19 for this equation.

21. In connection with Equation 4.4.16, suppose (y(0)=0) and (y'(0)=v_0), where (0

1. Let (T_1) be the time required for (y) to increase from zero to (y_{max}=2sin^{-1}(v_0/v_c)). Show that [{dyover dt}=sqrt{v_0^2-v_c^2sin^2y/2},quad 0le t
2. Separate variables in (A) and show that [T_1=int_0^{y_{max}}{duoversqrt{v_0^2-v_c^2sin^2u/2}} ag{B}]
3. Substitute (sin u/2=(v_0/v_c)sin heta) in (B) to obtain [T_1=2int_0^{pi/2}{d hetaoversqrt{v_c^2-v_0^2sin^2 heta}}. ag{C}]
4. Conclude from symmetry that the time required for ((y(t),v(t))) to traverse the trajectory [v^2=v_0^2-v_c^2sin^2y/2] is (T=4T_1), and that consequently (y(t+T)=y(t)) and (v(t+T)=v(t)); that is, the oscillation is periodic with period (T).
5. Show that if (v_0=v_c), the integral in (C) is improper and diverges to (infty). Conclude from this that (y(t)

22. Give a direct definition of an unstable equilibrium of (y''+p(y)=0).

23. Let (p) be continuous for all (y) and (p(0)=0). Suppose there’s a positive number ( ho) such that (p(y)>0) if (0

[alpha(r)=minleft{int_0^r p(x),dx, int_{-r}^0 |p(x)|,dx ight} mbox{quad and quad} eta(r)=maxleft{int_0^r p(x),dx, int_{-r}^0 |p(x)|,dx ight}.]

Let (y) be the solution of the initial value problem

and define (c(y_0,v_0)=v_0^2+2int_0^{y_0}p(x),dx).

1. Show that [0
2. Show that [v^2+2int_0^y p(x),dx=c(y_0,v_0),quad t>0.]
3. Conclude from (b) that if (c(y_0,v_0)<2alpha(r)) then (|y|0).
4. Given (epsilon>0), let (delta>0) be chosen so that [delta^2+2eta(delta)0), which implies that (overline y=0) is a stable equilibrium of (y''+p(y)=0).
5. Now let (p) be continuous for all (y) and (p(overline y)=0), where (overline y) is not necessarily zero. Suppose there’s a positive number ( ho) such that (p(y)>0) if (overline y

24. Let (p) be continuous for all (y).

1. Suppose (p(0)=0) and there’s a positive number ( ho) such that (p(y)<0) if (00). Conclude that (overline y=0) is an unstable equilibrium of (y''+p(y)=0).
2. Now let (p(overline y)=0), where (overline y) isn’t necessarily zero. Suppose there’s a positive number ( ho) such that (p(y)<0) if (overline y
3. Modify your proofs of (a) and (b) to show that if there’s a positive number ( ho) such that (p(y)>0) if (overline y- hole y

## Second order Autonomous Differential Equations

Introduce v = dR/dt. Then the differential equation is
v dv/dR = W^2 R.

Integrating once gives
v^2 - v0^2 = W^2 R^2 - W^2 R0^2
Where i have assumed v(t=0) = v0 and R(t=0) = R0.

a quick arrangement
v = +/- sqrt( v0^2 - W^2 R0^2 + W^2 R^2 )

and thus dR/dt = +/- sqrt( v0^2 - W^2 R0^2 + W^2 R^2 )

This is a seperable first order ODE

define a such that W^2 a^2 = v0^2 - W^2 R0^2

Then
dR/dt = +/- sqrt(W^2 a^2 + W^2 R^2)

The right hand side is +/- Wt.
To integrate the left hand side Put R = a*sinh(x) so that
dR = a cosh(x) dx then
sqrt( a^2 + R^2 ) = sqrt(a^2 + a^2 sinh^2(x)) = a sqrt(1 + sinh(x)^2)
= a sqrt(cosh^2(x)) = a*cosh(x).

This dR/sqrt(a^2 + R^2) -> dx
The integral is thus

arcsinh(R/a) - arcsinh(R0 / a) = +/- Wt
and therefore

R = a sinh( +/- Wt + arcsinh(R0 / a))
and a = sqrt(V0^2/W^2 - R0^2).

Plugging this in and checking shows us that the
- sign gives the v(0) = - v0

## Math 2552: Differential Equations - Fall 2018 Sec T

Recitations and TA office hours

 Recitation Section TA Name and Email TA Office Hours T1 MW 4:30-5:20pm Skiles 170 Alexander Winkles awinkles3 AT gatech.edu Thursday 11:15-12:15 Clough 280 (MathLab) T2 MW 4:30-5:20pm Skiles 255 Tao Yu tyu70 AT gatech.edu Thursday 3-4 Clough 280 (MathLab) T3 MW 4:30-5:20pm Skiles 257 Renyi Chen rchen342 AT gatech.edu Thursday 1:45-2:45pm Clough 280 (MathLab)

### Where to get help

• My Office Hours and your TA's office hours: see above. : Free service provided by the School of Math : Free service by GT. PLUS leader: Samantha Bordy, sbordy3 at gatech.edu. Session time and location: Tuesday and Thursday 6-7pm in CULC 262.
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### Announcements

Our final exam will be on Tuesday (12/11) 2:40-5:30pm, at Boggs B9. It is closed-book, closed-notes, and calculators are not allowed. You are allowed to bring one double-sided letter-size (8.5 by 11 inches) cheat sheet, which must be hand-written by yourself. (A printed or photocopied cheat sheet is not allowed.)
My office hours in this and next week will be: Wed (12/5) 3-4pm, Fri (12/7) 2-3pm, Mon (12/10) 11am-12pm.

## Elementary Differential Equations

Office : ACD 114A
Phone : (860) 405-9294
Office Hours : TTh 9:30 - 10:30am. and by appointment
Open Door Policy: You are welcome to drop by to discuss any aspect of the course, anytime, on the days I am on campus-- Tuesday, Thursday, and Friday.

MATH 2410 covers material mainly from chapters 1-5 of the textbook.

Class Meeting Times/Place: Tuesday, Thursday 2:00 - 3:15 p.m. Classroom ACD 206.

All classes start in ACD 206.

Elementary Differential Equations by William F. Trench.

It is an open source textbook available free online here

Homework is assigned every class and collected every Thursday. They are returned the following Tuesday with remarks and graded. The total weight of the homework grades is 50 point of the total 500 course points.

Exam Schedule: Exam 1: Tuesday, February 9 ,  2:00 - 3:15p.m., Room: ACD 206
Exam 2: Thursday, March 9 ,  2:00 - 3:15p.m., Room: ACD 206
Exam 3: Tuesday, April 11 ,  2:00 - 3:15p.m., Room: ACD 206
Final Exam: Tuesday, May 2, 1:30 - 3:30 p.m., Room: ACD 206

Grading Policy: Homework: 50, Exam 1: 100, Exam 2: 100, Exam 3: 100, Final Exam: 150.

 Date Chapter Topic Homework Week 1 Tues. 1/17 1.1 Applications leading to differential equations Thur. 1/19 1.2 Basic concepts Ch. 1.2: Exercises 1,2,4(a-d),5,7,9 Week 2 Tues. 1/24 1.3 Direction fields for first order ODEs Ch. 1.3: Exercises 1,2,3,4,5,12,13,14,15 Thur. 1/26 2.1 Linear first order equations Ch. 2.1 Exercises: 4,5,6,9,16,18,20,21 Week 3 Tues. 1/31 2.2 Separable equations Ch. 2.2 Exercises: 1,3,4,6,11,12,17,18 Thur. 2/2 2.3 Existence and uniqueness of solutions Ch. 2.3 Exercises: 1,2,3,4,14,16,17,20 Week 4 Tues. 2/7 Practice Exam 1 Practice Exam 1. Solutions Thur. 2/9 Snow Day! Week 5 Tues. 2/14 Exam 1 Thur. 2/16 3.1 Euler's Method. Ch. 3.1 Exercises: 1,4,6,14 Week 6 Tues. 2/21 4.1 Growth and decay Ch. 4.1 Exercises: 2,3,5,11 Thur. 2/23 4.2-4.3 Cooling, mixing and elementary mechanics Ch. 4.2 Exercises: 2,3,5,12 Ch. 4.3 Exercises:4,10 Week 7 Tues. 2/28 4.4 Autonomous second order equations Ch. 4.4 Exercises: Thur. 3/2 4.5 Applications to curves Ch. 4.5 Exercises: Week 8 Tues. 3/7 Review. Practice Exam 2 Practice Exam 2. Solutions Thur. 3/9 Exam 2 Week 9 Tues. 3/14 Spring recess Thur. 3/16 Spring recess Week 10 Tues. 3/21 5.1 Homogeneous linear equations Ch. 5.1 Exercises: Thur. 3/23 5.2 Constant coefficient homogeneous equations Ch. 5.2 Exercises: Week 11 Tues. 3/28 5.3 Nonhomogeneous linear equations Ch. 5.3 Exercises: Thur. 3/30 5.4 The method of underterminate coefficients 1 Ch. 5.4 Exercises: Week 12 Tues. 4/4 5.5 The method of underterminate coefficients 2 Ch. 5.5 Assignment 4 Week 12 Thur. 4/6 6.1-6.2 Spring problems Ch. 6.1 Week 12 Tues. 4/11 Review. Practice Exam 3 Practice Exam 3. Solutions Thur. 4/13 Exam 3 Week 14 Tues. 4/18 8.1-8.2 Laplace Transforms. Inverse transforms Ch. 8.1-8.2 Assignment 5 Thur. 4/20 8.3-8.4 Solutions to initial value problem Ch. 8.3 Exercises: Ch. 8.4 Exercises: Week 15 Tues. 4/25 8.5 Constant coefficient equations with piecewise continuous forcing functions Ch. 8.5 Exercises: Thur. 4/27 Review. Practice Final Exam Practice Final Exam. Solutions Week 16 Tues. 5/2 Final Exam, 1:30 p.m.-3:30 p.m.

This is linearizable by differentiation ode.

As you have seen, applied like that this substitution only increases the number of variables. You get a homogeneous pattern similar to an Euler-Cauchy equation by multiplying the original equation by $x^2$ , egin 0&=3(x^2y'')^2-2(3xy'+y)(x^2y'')+4(xy')^2 end One can now make this autonomous by borrowing the substitution $u(t)=y(e^t)$ from the Cauchy-Euler equation, $u'(t)=e^ty'(e^t)=xy'(x)$ , $u''(t)=e^<2t>y''(t)+e^ty'(e^t)=x^2y''(x)+u'(t)$ egin 0&=3(u''(t)-u'(t))^2-2(3u'(t)+u(t))(u''(t)-u'(t))+4u'(t)^2 &=3u''^2-6u''u'+3u'^2-6u''u'+6u'^2-2u''u+2u'u+4u'^2 &=3u''^2-12u''u'+13u'^2-2u''u+2u'u end As this is autonomous, one could insert $u'=v(u)$ , $u''=v'(u)v(u)$ . But even that does not appear helpful.

The expression can be rewritten as:

$3x^2(y'')^2 - 6 x y'y'' - 2yy'' + 4(y')^2 = 0.$

We make an observation as follows. Suppose that $y'' eq 0$ for all $x$ in the appropriate domain. The above equation simplifies to:

This implies that we can guess a possible solution could take the form of a simple polynomial $y(x) = C_0 + C_1 x + C_2 x^2 + C_3 x^3 ldots$ (This is due to the fact that the derivative of a polynomial is always one degree lower than that of the original polynomial.)

Recall that we have guessed that $y'' eq 0$ for all $x$ . This thus inspires that we guess $y'' = D$ for some constant $D$ . The equivalent "guess" for the solution would be

With that, substitute our guess into the main equation to obtain:

$3x^2(2C)^2 - 6 x(B+2Cx)(2C) = 2(A + Bx + Cx^2)(2C) - 4(B+2Cx)^2.$