How to Solve Coupled Linear Differential Equations

Written by selma leathem
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How to Solve Coupled Linear Differential Equations
A system of ordinary differential equations can be solved using matrix techniques. (maths mark image by Bram J. Meijer from

A common technique used to solve a system of coupled linear differential equations involves decoupling the equations through matrix methods and integrating each one separately. The key to the success of this method is the ability to decouple the equations by diagonalizing the square matrix which results when these equations are rewritten in matrix form. This technique requires matrix algebra, including eigenvalues and eigenvectors, and integral calculus, and there are many resources available on these topics such as Wolfram's Math World.

Skill level:


  1. 1

    Express the system of linear differential equations in matrix form. For example consider the following two differential equations

    dx/dt= axe + by (1)

    dy/dt= cx + dy (2)

    This can be rewritten in matrix form as dX/dt =Xdot = AX, where Xdot is a column matrix of the derivatives, A is 2 x 2 square matrix of the coefficients a,b,c and d, where a and b are in the first row and c and d the second, and X is a column matrix of the variables x and y. For more information as to how to write equations in matrix form see "Schaum's Outline of Theory and Problems of Matrix Operations": Richard Bronson: 1989.

  2. 2

    Calculate the eigenvalues by finding the solution to the characteristic equation for the matrix A. Eigenvalues are the characteristic roots of the characteristic equation, and the eigenvectors are the associated vectors. The characteristic equation is of the form det|A-LI|=0, where det is the determinant, L represents a matrix of eigenvalues and I is the identity matrix consisting only of elements with a value of one on its diagonal and zero elsewhere.

  3. 3

    Solve for the eigenvectors. Eigenvectors are related to the eigenvalues as follows:


    where S is a column matrix of eigenvectors.

  4. 4

    Diagonalize the matrix A by performing the following matrix operation:

    D = SA(inverse of S)

    where D is a matrix that only has values on its diagonal.

  5. 5

    Rewrite the original matrix equation dX/dt=Xdot= AX in terms of the diagonlized matrix of A by substituting X=SY and D = SA(inverse of S) to get


    This represents a system of decoupled differential equations.

  6. 6

    Integrate each row of the matrix equation dY/dt=Ydot=DY to find solutions for Y.

  7. 7

    Substitute the solution for Y back into the equation X=SY to get the solutions for the original equation.

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