Rewriting ODE's into lower orders

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In summary, the conversation discusses the correct approach for applying Euler's method in MATLAB to solve a system of ODEs. The example given involves a system with three variables, x, y, and z, and the equations x' = y, y' = z, and z' = xy - 2tz^2. The problem at hand involves a system with two variables, x and y, and the equations x' = y and y' = -sin(x). The conversation concludes with the confirmation that the same approach can be used for this problem.
  • #1
Euler2718
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Homework Statement



Express

[tex] \frac{d^{2}x}{dt^{2}} + \sin(x) = 0 [/tex]

In a system in terms of [itex]x'[/itex] and [itex]y'[/itex].

Homework Equations

The Attempt at a Solution


[/B]
I seen this example:

[tex]x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2} [/tex]

Where they then wrote:

[tex] x^{\prime} = y [/tex]
[tex] y^{\prime} = z [/tex]
[tex] z^{\prime} = y^{\prime\prime} = x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2} [/tex]
So:
[tex] z^{\prime} = xy - 2tz^{2} [/tex]
Since [itex]y^{\prime\prime} = x^{\prime} = z[/itex]

And thus an Euler's method can be devised in MATLAB, given some IVP's of course. Is this then the correct approach for my problem:

[tex] x^{\prime} = y [/tex]
[tex] y^{\prime} = x^{\prime\prime} = -\sin(x) [/tex]

Not really familiar with ODE's and such processes, but I need to apply this correctly to proceed with my work.
 
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  • #2
Morgan Chafe said:

Homework Statement



Express

[tex] \frac{d^{2}x}{dt^{2}} + \sin(x) = 0 [/tex]

In a system in terms of [itex]x'[/itex] and [itex]y'[/itex].

Homework Equations

The Attempt at a Solution


[/B]
I seen this example:

[tex]x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2} [/tex]

Where they then wrote:

[tex] x^{\prime} = y [/tex]
[tex] y^{\prime} = z [/tex]
[tex] z^{\prime} = y^{\prime\prime} = x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2} [/tex]
So:
[tex] z^{\prime} = xy - 2tz^{2} [/tex]
Since [itex]y^{\prime\prime} = x^{\prime} = z[/itex]

And thus an Euler's method can be devised in MATLAB, given some IVP's of course. Is this then the correct approach for my problem:

[tex] x^{\prime} = y [/tex]
[tex] y^{\prime} = x^{\prime\prime} = -\sin(x) [/tex]
This is it right here.
 
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  • #3
Mark44 said:
This is it right here.

Thanks. I really had to be sure before continuing.
 

Related to Rewriting ODE's into lower orders

1. What is the purpose of rewriting ODE's into lower orders?

The purpose of rewriting ODE's (ordinary differential equations) into lower orders is to simplify the equation and make it easier to solve. This is especially useful when dealing with complex equations that involve multiple variables and functions.

2. How do you rewrite an ODE into lower orders?

To rewrite an ODE into lower orders, you can use the substitution method or the elimination method. The substitution method involves substituting a new variable for the highest derivative in the equation, while the elimination method involves eliminating derivatives by expressing them in terms of lower order derivatives.

3. What are the benefits of rewriting ODE's into lower orders?

Rewriting ODE's into lower orders can help reduce the complexity of the equation, making it easier to understand and solve. It can also help identify patterns and relationships between variables, which can provide insight into the behavior of the system described by the equation.

4. Are there any limitations to rewriting ODE's into lower orders?

While rewriting ODE's into lower orders can be helpful in many cases, it may not always be possible or practical. Some equations may not have a simple way of being rewritten, and in some cases, the resulting equation may be even more complicated than the original one.

5. Can rewriting ODE's into lower orders be applied to all types of differential equations?

No, rewriting ODE's into lower orders is typically only applicable to linear differential equations. Nonlinear equations may not have a simple way of being rewritten, and the resulting equation may not be easier to solve. Additionally, some special types of ODE's, such as those with variable coefficients, may also be difficult to rewrite into lower orders.

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