Differential equation periodic and non periodic solutions.

In summary: T}\int_{0}^{T}a(s)ds=\frac{1}{T}\int_{0}^{t+T}a(s)ds-\frac{1}{T}\int_{0}^{t}a(s)ds=0.$$Thus, if $x(t)$ is periodic, then $\bar{a}=0$.($\Leftarrow$) Now assume that $\bar{a}=0$. Then we have $\frac{1}{T}\int_{0}^{T}a(s)ds=0$. This implies that $e^{\int_{0}^{t+T}a(s)ds-\int_{0}^{t}a(s)ds}=e^{\int
  • #1
ELESSAR TELKONT
44
0
I have the next problem. Let be the linear differential equation [tex]\dot x=a(t)x[/tex] where [tex]a(t)[/tex] is a periodic function of fundamental period [tex]T>0[/tex],

a) Give two examples: where the general solution is a periodic function again and where the general solution is a non-periodic function.

b) Show that the general solution is periodic of fundamental period [tex]T>0[/tex] iff [tex]\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0[/tex].


I have done yet the case when differential equation is [tex]\dot x=a(t)[/tex] and I have proof that solutions to this one are periodic iff [tex]\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0[/tex]. Can I use this fact?

I also know that the general solution to the equation of my question is [tex]x(t)=ce^{\int_{}^{}a(s)ds}[/tex] and that the solutions to every IVP (except for that points over the equilibrium solution [tex]x(t)\equiv 0[/tex]) are [tex]x(t)=x_{0}e^{\int_{t_{0}}^{t}a(s)ds}[/tex] and that it implies that if this is periodic [tex]x_{0}e^{\int_{t_{0}}^{t}a(s)ds}-x_{0}e^{\int_{t_{0}}^{t+T}a(s)ds}=0[/tex]

and [tex]x_{0}e^{\int_{t_{0}}^{t}a(s)ds-\int_{t_{0}}^{t+T}a(s)ds}=x_{0}e^{\int_{t_{0}}^{t}a(s)ds-\int_{t_{0}}^{t}a(s)ds-\int_{t}^{t+T}a(s)ds}=x_{0}e^{-\int_{t}^{t+T}a(s)ds}=0[/tex]

but here I'm lost because it is that the integral couldn't exist but if [tex]a(t)[/tex] is periodic and that implies that integral exists. How proof that, and what functions are exaples of each type.
 
Physics news on Phys.org
  • #2
A) To give an example of a periodic solution, consider the differential equation $\dot{x}=\sin(t)x$. This equation has a general solution $x(t)=Ce^{\int_{0}^{t}\sin(s)ds}=Ce^{-\cos(t)+C_1}$. Since the integrand $\sin(t)$ is periodic with period $2\pi$, then $Ce^{-\cos(t)+C_1}$ is also periodic with period $2\pi$.To give an example of a non-periodic solution, consider the differential equation $\dot{x}=t^2x$. This equation has a general solution $x(t)=Ce^{\int_{0}^{t}s^2ds}=Ce^{\frac{t^3}{3}+C_1}$. Since the integrand $t^2$ is not periodic, then $\frac{t^3}{3}+C_1$ is not periodic and thus the general solution $x(t)=Ce^{\frac{t^3}{3}+C_1}$ is also not periodic.B) Let $T>0$ be the fundamental period of the function $a(t)$. We will show that the general solution $x(t)=Ce^{\int_{0}^{t}a(s)ds}$ is periodic of period $T$ if and only if $\bar{a}=\frac{1}{T}\int_{0}^{T}a(s)ds=0$.($\Rightarrow$) Assume that $x(t)$ is periodic of period $T$. Then we have $x(t+T)=x(t)$ for all $t$. This implies that $Ce^{\int_{0}^{t+T}a(s)ds}=Ce^{\int_{0}^{t}a(s)ds}$, or equivalently, $e^{\int_{0}^{t+T}a(s)ds-\int_{0}^{t}a(s)ds}=1$. Taking the average over the period $T$, we obtain $$\bar{a}=\frac{1
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It is used to model various physical phenomena in fields such as physics, engineering, and economics.

2. What is a periodic solution?

A periodic solution to a differential equation is a solution that repeats itself after a certain period of time or distance. This means that the function will have the same values at regular intervals.

3. How is a periodic solution different from a non-periodic solution?

A non-periodic solution to a differential equation does not repeat itself after a certain period of time or distance. This means that the function will have different values at different intervals and does not exhibit any repeating patterns.

4. Can a differential equation have both periodic and non-periodic solutions?

Yes, a differential equation can have both periodic and non-periodic solutions. It depends on the initial conditions and the parameters of the equation.

5. How are differential equations used in real-world applications?

Differential equations are used in a wide range of applications, including physics, engineering, biology, and economics. They are used to model and predict the behavior of complex systems and help us understand and solve real-world problems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
520
  • Calculus and Beyond Homework Help
Replies
5
Views
846
  • Calculus and Beyond Homework Help
Replies
31
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
646
  • Calculus and Beyond Homework Help
Replies
4
Views
533
  • Calculus and Beyond Homework Help
Replies
7
Views
80
  • Calculus and Beyond Homework Help
Replies
6
Views
88
  • Calculus and Beyond Homework Help
Replies
2
Views
133
Back
Top