Question

For the equations : x′′=3x−y, y′=x+9y

a) write an equivalent first order system.
b) determine the dimension of the system.
c) State whether or not it is linear, and if linear, state whether or not it is homogeneous.

Answer 1

The system of equations given is transformed into an equivalent first order system by introducing new variables. The dimension of this system is 3, and it is both linear and homogeneous.

Step-by-step explanation:

To answer the question regarding the equations: x''=3x-y and y'=x+9y:

1. Equivalent first order system: We introduce two new variables, u and v, where u = x' and v = y. Thus we can rewrite the equations as:
   • u' = 3x - v (since x'' = u')
   • u = x'
   • v' = x + 9v (since y' = v')
   The equivalent first order system is then:
   • x' = u
   • u' = 3x - v
   • v' = x + 9v
2. Dimension of the system: The system has three first order ordinary differential equations, so its dimension is 3.
3. Nature of the system: The system is linear because it can be written in the form AX = B, where A is a matrix of coefficients, X is a vector of the functions we are looking for (x, u, v), and B is a vector (here B would be the zero vector since the system does not have any terms that are independent of x, u, or v). It is also homogeneous because there are no constant terms or functions of time alone on the right side of the system of equations; all terms involve the dependent variables x, u, or v.