Solution for system of diff equations

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In summary, a system of first order differential equations can be converted into a single second order differential equation and vice versa. However, this method may not always be simpler than computing eigenvalues and eigenvectors of a matrix and may require choosing independent boundary conditions for each equation.
  • #1
Jhenrique
685
4
Given system like ##\begin{bmatrix}
x'(t)\\
y'(t)\\
\end{bmatrix}

=

\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}

\begin{bmatrix}
x(t)\\
y(t)\\
\end{bmatrix}## can be rewritten as ##(-1)u''(t) + (bd)u'(t) + (ad+c)u(t) = 0## with ##\begin{bmatrix}
x(t)\\
y(t)\\
\end{bmatrix}

=

\begin{bmatrix}
u(t)\\
u'(t)\\
\end{bmatrix}##

In other words, if a diff equation of 2nd order can be disassembled in a system of 1nd order, thus a system of 1nd order can be assembled in a diff equation of 2nd order, correct? This way, I don't need to find the eigenvalues and eigenvectors...
 
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  • #2
How do you guarantee that ##y(t)=u'(t)=x'(t)## will always work? I don't see why this would be so... they look independent to me in the first equation...
 
  • #3
Yes, that is true. Dropping the matrix notation, if x'= ax+ by and y'= cx+ dy. then x''= ax'+ by'= ax'+ b(cx+ dy)= ax'+ bcx+ bdy. From the first equation, by= x'- ax so that
x''= ax'+ bcx+ d(x'- ax)= (a+ d)x'+ (bc- ad)x. We can always convert a system of n first order differential equations to a single nth order differential equation.
 
  • #4
HallsofIvy said:
Yes, that is true. Dropping the matrix notation, if x'= ax+ by and y'= cx+ dy. then x''= ax'+ by'= ax'+ b(cx+ dy)= ax'+ bcx+ bdy. From the first equation, by= x'- ax so that
x''= ax'+ bcx+ d(x'- ax)= (a+ d)x'+ (bc- ad)x. We can always convert a system of n first order differential equations to a single nth order differential equation.

And this method isn't more simple than compute the eigenvalues and eigenvectors of a matrix?
 
  • #5
Matterwave has seen the problem with the OP's idea.

You can convert the two equations into second order equations
x''= (a+d)x'+ (bc-ad)x
y''= (a+d)y'+ (bc-ad)y

But it doesn't follow that y = x'. You can choose two independent boundary conditions for each equation.

If it did follow that y = x', it would also follow that x = y', and that clearly means "something is wrong here".

In fact the OP's idea is used in reverse: to solve a single differential equation of order n, it is often easiest to convert it into a system of n first-order equations.
 

1. What is a system of differential equations?

A system of differential equations is a set of equations that describes the relationship between multiple variables and their rates of change over time. These equations are typically used to model complex systems in physics, engineering, and other scientific fields.

2. What is a solution for a system of differential equations?

A solution for a system of differential equations is a set of values for the variables that satisfies all of the equations in the system. This means that when the values are plugged into the equations, they produce a consistent set of results that accurately represent the behavior of the system over time.

3. How do you solve a system of differential equations?

There are several methods for solving a system of differential equations, including analytical methods (such as separation of variables and substitution) and numerical methods (such as Euler's method and Runge-Kutta methods). The method used will depend on the complexity of the system and the desired level of accuracy.

4. What are the applications of solving systems of differential equations?

Solving systems of differential equations has many real-world applications, such as predicting the motion of objects in physics, modeling population growth in biology, and simulating chemical reactions in chemistry. It is also used in engineering to design and optimize systems such as circuits, control systems, and chemical processes.

5. Can a system of differential equations have multiple solutions?

Yes, a system of differential equations can have multiple solutions. This can happen when the system is nonlinear or when there are multiple sets of initial conditions that can produce different solutions. In some cases, these solutions may represent different behaviors or outcomes of the system.

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