Second-order Coupled O.D.E.s with constant coefficients

• ab959
In summary, the conversation is about solving a system of two coupled ODEs and finding an analytic solution. The person has attempted different methods, including a Laplace transform and representing the equations as a system of first order equations, but has not been successful. They are considering using an arbitrary function and exploiting symmetry to find a solution. Another person suggests solving the first equation for U_2 and plugging it into the second equation. There is a question about the presence of "\partial x^2" in the second term of the left hand side of each equation.
ab959

Homework Statement

I am trying to solve a system of two coupled ODEs. I am interested in an analytic solution if that is possible. I know it will be messy.

$$\frac{\partial^2 U_1}{\partial x^2}+a_1\frac{\partial U_1}{\partial x^2}+b_1 U_1 = c_1 U_2$$

$$\frac{\partial^2 U_2}{\partial x^2}+a_2\frac{\partial U_2}{\partial x^2}+b_2 U_2 = c_2 U_1$$

The Attempt at a Solution

I have attempted a few methods although I haven't pursued them too hard as I don't want to head down the wrong path. I found that a Laplace transform was too hard to invert. I tried to represent it as a system of 4 first order equations however the inversion was very very messy. Finally, the method I was thinking of employing was to treat the U_2 in the first equation as an arbitrary function and then exploiting the symmetry to find a solution of U_1/U_2 in terms of something in the form of:

$$U_{1,2}=... + \int^x_{x_0} U_{2,1}(\eta)e^{(x-\eta)k}d\eta$$

, where k is a variable in terms of a_1 and b_2. I do not know if this will work and have not done much in this area.

Does anyone have any suggestions as I am beginning to feel a bit out of my depth. Thanks.

Solve the first equation for $U_2$ and plug it into the second equation.

ab959 said:
$$\frac{\partial^2 U_1}{\partial x^2}+a_1\frac{\partial U_1}{\partial x^2}+b_1 U_1 = c_1 U_2$$

$$\frac{\partial^2 U_2}{\partial x^2}+a_2\frac{\partial U_2}{\partial x^2}+b_2 U_2 = c_2 U_1$$

Why is there a "$\partial x^2$" in the second term of the left hand side of each equation?

What are second-order coupled O.D.E.s with constant coefficients?

Second-order coupled O.D.E.s (ordinary differential equations) with constant coefficients are equations that involve two unknown functions and their derivatives, where the coefficients of the derivatives are constants (i.e. they do not depend on the independent variable).

What is the difference between first-order and second-order coupled O.D.E.s?

The main difference is the number of unknown functions and their derivatives involved. First-order coupled O.D.E.s involve one unknown function and its derivative, while second-order coupled O.D.E.s involve two unknown functions and their derivatives.

How do you solve second-order coupled O.D.E.s with constant coefficients?

The general approach to solving these types of equations is to first convert them into a system of first-order equations by introducing new variables. Then, techniques such as substitution, elimination, and matrix methods can be used to solve the system of equations.

What are some real-world applications of second-order coupled O.D.E.s with constant coefficients?

These types of equations are commonly used in physics and engineering to model systems that involve two interacting variables. For example, they can be used to describe the motion of a pendulum or the behavior of an electric circuit.

Are there any special cases of second-order coupled O.D.E.s with constant coefficients?

Yes, there are a few special cases that have simpler solutions. These include equations with equal coefficients (known as homogeneous equations) and equations with one constant coefficient equal to zero (known as linearly independent equations).

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