Second order ordinary differential equation to a system of first order

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• LSMOG
In summary, the conversation discusses the process of converting a second order ordinary differential equation to a system of first order differential equations and writing it in matrix form. The speaker took this process from a book on optimal control by LM Hocking, but their solution differs from the one in the book. However, there is nothing wrong with the speaker's approach and it still leads to the same solution. The only difference is in the definition of ##x_2##. The speaker's method requires adjusting constants more, while the book's method gives a specific form. The speaker thanks the other person for their input.
LSMOG
I tried to convert the second order ordinary differential equation to a system of first order differential equations and to write it in a matrix form. I took it from the book by LM Hocking on (Optimal control). What did I do wrong in this attachment because mine
differs from the book?. I've attached both the book solution and mine. Thanks.

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LSMOG said:
I tried to convert the second order ordinary differential equation to a system of first order differential equations and to write it in a matrix form. I took it from the book by LM Hocking on (Optimal control). What did I do wrong in this attachment because mineView attachment 226158 differs from the book?. I've attached both the book solution and mine. Thanks.
I don't think there is anything wrong with your way (except for the ##\frac 1 k## you have penciled in front of the matrix in your equation). It still leads to the same solution to the differential equation. Your way does require fiddling with the constants a little more to get to that solution, which may be why your textbook gives the particular form you found there.

You didn't do anything wrong. Just that your definition of ##x_2## is different from that in the book.
When I do this I do it the same way you do, but perhaps the book author has some specific reason for his approach ?

Thanks very much

1. How is a second-order ordinary differential equation converted into a system of first-order equations?

To convert a second-order ordinary differential equation into a system of first-order equations, we introduce new variables and rewrite the original equation as a set of first-order equations. This is typically done by setting one of the variables equal to its derivative and introducing another variable to represent the second derivative.

2. What are the advantages of expressing a second-order ordinary differential equation as a system of first-order equations?

By converting a second-order ordinary differential equation into a system of first-order equations, we can use numerical methods that are specifically designed for solving systems of equations. This allows for more accurate and efficient solutions to complex problems.

3. Can all second-order ordinary differential equations be converted into a system of first-order equations?

Yes, any second-order ordinary differential equation can be rewritten as a system of first-order equations. However, the resulting system may be more complex and difficult to solve than the original equation.

4. How does solving a system of first-order equations differ from solving a second-order ordinary differential equation?

Solving a system of first-order equations involves finding the values of each variable at discrete points in time, while solving a second-order ordinary differential equation involves finding a function that satisfies the equation at all points in time. Additionally, numerical methods are typically used to solve systems of equations, while analytical methods can sometimes be used for second-order differential equations.

5. Are there any practical applications for converting a second-order ordinary differential equation into a system of first-order equations?

Yes, there are many practical applications for this conversion. It is commonly used in physics and engineering to model complex systems, such as in the study of fluid dynamics or electrical circuits. It is also used in numerical analysis to solve differential equations numerically.

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