Are There Complex Solutions for a 2nd Order ODE?

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In summary: So, u and v are two linearly independent solutions, but there are actually infinitely many solutions of the form u+iv.In summary, the conversation discusses a constant coefficients ODE and its solutions, which can be written in terms of real values or complex values. The proof presented shows that the real and imaginary parts of a complex solution are also solutions to the ODE. The conversation also clarifies that a 2nd order ODE has a general 2 parameter family of solutions, with u and v being two linearly independent solutions. However, there are infinitely many solutions of the form u+iv.
  • #1
ognik
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Hi - I have y''+9y=0 which is a constant coefficients ODE, the CE is then $r^2+9=0$ and I get a general solution $ y=C_1e^{3ix}+C_2e^{-3ix} $

But I have seem these solutions written as ACos3x+BSin3x. If I use Euler on my solution, I get $ C_1(Cos3x + iSin3x) +C_2(Cos3x-iSin3x) $ ... Are they just using $ A=C_1+C_2 $ and $ B=i(C_1-C_2) $ or am I missing something?
 
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Suppose we are given the ODE:

\(\displaystyle ay''+by'+cy=0\tag{1}\)

Let $z(x)=u(x)+iv(x)$ be a solution to (1), where $a$, $b$, and $c$ are real numbers. Then the real part $u(x)$ and the imaginary part $v(x)$ are real valued solutions of (1).

Proof:

By assumption $az''+bz'+cz=0$, and hence:

\(\displaystyle a(u''+iv'')+b(u'+uv')+c(u+iv)=0\)

Which we can arrange as:

\(\displaystyle (au''+bu'+cu)+i(av''+bv'+cv)=0\)

A complex number is zero iff both its real and imaginary parts are zero. Thus, we must have:

\(\displaystyle au''+bu'+cu=0\)

\(\displaystyle av''+bv'+cv=0\)

And this means that both $u(x)$ and $v(x)$ are real-valued solutions of (1).
 
  • #3
Thanks, very clear. Question - A 2nd order ODE has at most 2 solutions, so are u an v those 2 solutions, or are there other complex solutions (besides u+iv)?
 
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  • #4
ognik said:
Hi - I have y''+9y=0 which is a constant coefficients ODE, the CE is then $r^2+9=0$ and I get a general solution $ y=C_1e^{3ix}+C_2e^{-3ix} $

But I have seem these solutions written as ACos3x+BSin3x. If I use Euler on my solution, I get $ C_1(Cos3x + iSin3x) +C_2(Cos3x-iSin3x) $ ... Are they just using $ A=C_1+C_2 $ and $ B=i(C_1-C_2) $ or am I missing something?

Yes that is exactly what they are doing. Even complex constants are still constants.
 
  • #5
ognik said:
Thanks, very clear. Question - A 2nd order ODE has at most 2 solutions, so are u an v those 2 solutions, or are there other complex solutions (besides u=iv)?

A 2nd order ODE will have a general 2 parameter family of solutions. For example, consider the homogeneous ODE:

\(\displaystyle ay''+by'+cy=0\)

Let $r_1$ and $r_2$ be the roots of the associated characteristic equation. Then, we know, by the principle of superposition that the general solution is:

\(\displaystyle y(x)=c_1e^{r_1x}+c_2e^{r_2x}\)

Since there are an infinite number of ways we can choose the two parameters, there are an infinite number of solutions. What we do, is identify 2 linearly independent solutions using the roots of the characteristic equation, and then affix a parameter to each to get the general solution.
 

FAQ: Are There Complex Solutions for a 2nd Order ODE?

What is an ODE?

An ODE (Ordinary Differential Equation) is a mathematical equation that describes how a variable changes over time. It involves the derivative of a function and can be used to model a wide range of physical phenomena.

Why is it important to write ODE solutions as reals?

Writing ODE solutions as reals means that the solutions are expressed as real numbers, rather than complex numbers. This is important because real numbers are more easily understood and can be interpreted in a physical context.

How do you write an ODE solution as reals?

To write an ODE solution as reals, you need to use techniques such as separation of variables, substitution, or integrating factors. These techniques allow you to manipulate the ODE into a form where the solution can be expressed as a real-valued function.

What are the benefits of writing ODE solutions as reals?

Writing ODE solutions as reals allows for a more intuitive understanding of the solution and its behavior. It also simplifies the solution and makes it easier to analyze and interpret in a real-world context.

Are there any limitations to writing ODE solutions as reals?

Yes, there are some ODEs that cannot be solved using real numbers and require complex numbers to find a solution. In these cases, writing the solution as reals may not be possible or may result in an incorrect solution. Additionally, some ODEs may have multiple real solutions, so it is important to check for all possible solutions when writing the ODE solution as reals.

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