Linear System Example: Repeated Real Root

We solve Example 2 of §56 in [Simmons, Second edition], on p. 431–2,

\{ \begin{matrix}\frac{dx}{dt} & = & 3x − 4y \\ \frac{dy}{dt} & = & x − y \end{matrix}

by an alternate method. We apply the Laplace transform with the generic initial conditions x(0) = k_{1} and y(0) = k_{2} to get

\{ \begin{matrix}pX − k_{1} & = & 3X − 4Y \\ pY − k_{2} & = & X − Y \end{matrix} or \{ \begin{matrix}(3 − p)X − 4Y & = & −k_{1} \\ X − (1 + p)Y & = & −k_{2}. \end{matrix}

We solve these simultaneous linear equations for X and Y by elimination,

\{ \begin{matrix}[4 · 1 − (1 + p)(3 − p)]X & = & −4k_{2} + (1 + p)k_{1} & = & (p + 1)k_{1} − 4k_{2} \\ [−(3 − p)(1 + p) + 1 · 4]Y & = & −(3 − p)k_{2} + k_{1} & = & k_{1} + (p − 3)k_{2}. \end{matrix}

The expressions in [ ] are both

4 − (1 + p)(3 − p) = 4 − (3 + 2p − p^{2}) = p^{2} − 2p + 1 = (p − 1)^{2}.

We therefore divide out by this and express everything in terms of p − 1, with an eye towards using the shift formula for Laplace transforms,

\{ \begin{matrix}X & = & \frac{(p + 1)k_{1} − 4k_{2}}{(p − 1)^{2}} & = & \frac{(p − 1)k_{1} + 2k_{1} − 4k_{2}}{(p − 1)^{2}} & = & \frac{k_{1}}{p − 1} + \frac{2k_{1} − 4k_{2}}{(p − 1)^{2}} \\ Y & = & \frac{k_{1} + (p − 3)k_{2}}{(p − 1)^{2}} & = & \frac{(p − 1)k_{2} + k_{1} − 2k_{2}}{(p − 1)^{2}} & = & \frac{k_{2}}{p − 1} + \frac{k_{1} − 2k_{2}}{(p − 1)^{2}} \end{matrix}

Finally, we apply the inverse Laplace transform and find

\{ \begin{matrix}x & = & k_{1}e^{t} + (2k_{1} − 4k_{2})te^{t} & = & [k_{1} + (2k_{1} − 4k_{2})t]e^{t} \\ y & = & k_{2}e^{t} + (k_{1} − 2k_{2})te^{t} & = & [k_{2} + (k_{1} − 2k_{2})t]e^{t}. \end{matrix}

This yields the solution (23) in the book if we take k_{1} = 2 and k_{2} = 1, and the solution (25) if we take k_{1} = 1 and k_{2} = 0. The correct values of k_{1} and k_{2} to use are obvious from the initial conditions. Moreover, we recover (26) if we put c_{1} = k_{2} and c_{2} = k_{1} − 2k_{2}.

Compare this treatment with the traditional one in the book to decide which is shorter or easier or more efficient. Draw your own conclusions.

110.302 Differential Equations JMB File: lsysrep, Rev. A; 30 Aug 2001; Page 1