Coupled Differential Equations (DP IB Applications & Interpretation (AI): HL): Revision Note

Solving coupled differential equations

How do I write a system of coupled differential equations in matrix form?

• The coupled differential equations considered in this part of the course will be of the form

  $\frac{dx}{dt}=ax+by$

  $\frac{dy}{dt}=cx+dy$

  •  $a,b,c,d\in \mathrm{ℝ}$ are constants whose precise value will depend on the situation being modelled

• This system of equations can also be represented in matrix form:

$\left(\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

• You can use the dot notation for the derivatives:

$\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

• This can be written even more succinctly as $\dot{\mathit{x}}\mathbf{=}\mathit{M}\mathit{x}$

  • $\dot{\mathit{x}}=\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)$

  • $\mathit{M}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$

  • $\mathit{x}=\left(\begin{array}{c}x\\ y\end{array}\right)$

• For example,

  • $\frac{dx}{dt}=2x-3y$ and $\frac{dy}{dt}=x+4y$ can be modelled as

  • $\left(\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right)=\left(\begin{array}{cc}2& -3\\ 1& 4\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$

How do I find the exact solution for a system of coupled differential equations?

• The exact solution of the coupled system $\dot{\mathit{x}}=\mathit{M}\mathit{x}$ depends on the eigenvalues and eigenvectors of the matrix of coefficients 

• Suppose for matrix $\mathit{M}$

  • ${\lambda }_1$ and ${\lambda }_2$ are the eigenvalues

  • ${\mathit{p}}_1$ and ${\mathit{p}}_2$ are corresponding eigenvectors respectively

• The exact solution to the system of coupled differential equations is then
  

  $\mathit{x}=A{\mathrm{e}}^{{\lambda }_{1}t}{\mathit{p}}_1+B{\mathrm{e}}^{{\lambda }_{2}t}{\mathit{p}}_{\mathbf{2}}$

• If initial or boundary conditions have been provided you can use these to find the precise values of the constants $A$ and $B$

  • Finding the values of $A$ and $B$ will generally involve solving a set of simultaneous linear equations