Seeking advice about solving an ODE

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In summary, the ODE describing the dependence of y on B can be solved for y_0, y_1, y_2, etc. by an integrating factor. However, only the odd terms contribute.
  • #1
Hello! I am trying to solve the time dependent Schrodinger equation for a 2x2 system and I ended up with this ODE:

$$y''=-iA\sin{(\omega t)}y'-B^2y$$

with the initial conditions ##y(t=0)=0## and ##y'(t=0)=B##. I can look at it numerically but I was wondering if there is a way to get something analytical out of it. In my case I have ##B<<A,\omega## (not sure if that helps), but I am interested in the way ##y## depends on B, so I can't just drop that term either. Any advice would be greatly appreciated. Thank you!
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  • #2
First set [itex]\tau = \omega t[/itex] so that [tex]
\frac{d^2 y}{d\tau^2} + i\alpha\sin(\tau) \frac{dy}{d\tau} + \epsilon^2 y = 0[/tex] subject to [itex]y(0) = 0[/itex], [itex]y'(0) = \epsilon[/itex] where [itex]\alpha = A/\omega[/itex] and [itex]\epsilon = B/\omega \ll 1[/itex].

Pose an asymptotic expansion [tex]
y(t) \sim \sum_{k=0}^\infty \epsilon^ky_k(t)[/tex] in the limit [itex]\epsilon \to 0[/itex]. Then the ODE becomes [tex]
(y''_0 + i\alpha\sin(\tau) y'_0) + \epsilon(y''_1 + i\alpha\sin(\tau) y'_1) \\
+ \sum_{k=2}^\infty \epsilon^k \left(y''_k + i\alpha\sin(\tau)y'_k + y_{k-2}\right) = 0[/tex] so that considering coefficients of [itex]\epsilon^k[/itex] we have [tex]
y''_0 + i\alpha \sin(\tau) y_0' &= 0 \\
y''_1 + i\alpha \sin(\tau) y_1' &= 0 \\
y''_k + i\alpha \sin(\tau) y_k' &= -y_{k-2} \end{split}[/tex] subject to the initial conditions
[tex]\begin{gather*}y_k(0) = y'_k(0) = 0, \quad k \neq 1 \\
y_1(0) = 0, \quad y_1'(0) = 1.\end{gather*}[/tex] The equation for [itex]y_k[/itex] can be solved by an integrating factor once [itex]y_0, \dots, y_{k-1}[/itex] are known. ("Solved" in the sense that the solution can be written in terms of integrals, but the integrals may or may be expressible in terms of elementary functions.) It seems that [itex]y_{2k} \equiv 0[/itex] so only the odd terms contribute.
Note that this expansion is only valid so long as [tex]
\left| \frac{y_{2k+3}}{y_{2k+1}} \right| < \frac 1{\epsilon^2}.[/tex]
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  • #3
If we omit ##-B^2 y## in RHS, we can solve the simplified ODE,
[tex]y=B\int_0^t du \ e^{\frac{iA}{\omega}\cos \omega u} [/tex]
We may be able to expect that in a short time the solution of the original ODE does not so much different from it. I would appreciate it If you could check the difference with your numerical solution.
  • #4
It i spossible to obtain a power series solution. The system is of the form [tex]
\dot x = M(t)x[/tex] for [itex]x \in \mathbb{R}^2[/itex] where [itex]x = (y, \dot y)^T[/itex] and [tex]
M = \begin{pmatrix} 0 & 1 \\ -B^2 & -iA\sin \omega t \end{pmatrix}.[/tex] Then setting [tex]\begin{split}
x(t) &= \sum_{n=0}^\infty a_nt^n \\
M(t) &= \sum_{n=0}^\infty M_nt^n \end{split}[/tex] we have [tex]
\sum_{n=0}^\infty (n+1)a_{n+1}t^n = \sum_{n=0}^\infty t^n \left[
\sum_{m=0}^n M_{n-m}a_m
\right][/tex] whence [tex]
a_{n+1} = \frac{1}{n+1} \sum_{m=0}^n M_{n-m} a_m, \qquad n \geq 0
[/tex] with [itex]a_0 = (0, B)^T[/itex].

What is an ODE?

An ODE (ordinary differential equation) is a mathematical equation that involves derivatives of one or more dependent variables with respect to one or more independent variables. It is commonly used in physics, engineering, and other scientific fields to model various physical phenomena.

Why do we need to solve ODEs?

ODEs are important because they allow us to describe and analyze real-world systems and predict their behavior over time. They are used in a wide range of applications, such as predicting the trajectory of a projectile, modeling population growth, or understanding the dynamics of chemical reactions.

What methods are commonly used to solve ODEs?

There are several methods for solving ODEs, including analytical methods (such as separation of variables or integrating factors) and numerical methods (such as Euler's method or Runge-Kutta methods). The choice of method depends on the complexity of the ODE and the desired level of accuracy.

How can I know if my solution to an ODE is correct?

One way to check the correctness of a solution to an ODE is to substitute it back into the original equation and see if it satisfies the equation. Another method is to compare the solution to known solutions or use a computer program to verify the solution.

Is there a general approach for solving ODEs?

Yes, there are general steps that can be followed for solving ODEs. First, identify the type of ODE (e.g. linear, separable, etc.) and choose an appropriate method. Then, solve the ODE using the chosen method and check the solution for correctness. Finally, apply any initial or boundary conditions to determine the specific solution for the given problem.

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