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Using Matrices to Solve Differential Equations?

TopWe can use matrices to solve a system of differential equation. In mathematics Differential Equations are nothing but Derivatives of Linear Equations. Let us consider an example in which we depict process of Using Matrices to solve differential equations. Suppose we have two differential equations given as:
dx / dt = 1x - 2y equation (1)
dy / dt = 4x + 4y equation (2)
In matrix form we can write these equations as:

dx / dt

=

1

-2

*

X

dy / dt

 

4

4

 

Y

 
Where, 'A' is the coefficient matrix given as:

1

-2

4

4

And 'X' is column matrix:

X

Y

Next we use the characteristic equation given as determinant |A - LI| = 0 to get:
Where, I =
 

1

0

0

-1


1 – L

- 2

=

0

4

4 – L

 

 

Solving the determinant we get:
(1 – L) (4 – L) – (4) (2) = 0,
Or 4 – L + L2 – 4L – (- 8) = 0,
OR L2 – 5L + 4 = 0,
OR L = 4 and L = 1,
Next we calculate eigen vectors on basis of values of 'L' using the formula: A P = L P. Where, 'P' is a column matrix of eigen vectors. Diagonalize the matrix 'A' by performing following matrix operation: D = P A (inverse of P).
Where, 'D' is a matrix that only has values on its diagonal. Rewrite the matrix 1 by substituting X = PY and D = PA (inverse of P) to get:
dY / dt = DY equation 1
Last step is to find out integral of every row of matrix equation 1 to find solutions for 'Y'. Substitute the value of 'Y' in equation X = PY to get solutions for original equation. This is how you solve system of coupled differential equations using matrix representation.