ordinary differential equations
I desperately need to solve a coupled system of linear second order differential equations of the form:
$$x''+ax'+bx-cy'-dy=0$$
$$y''+ay'+by+cx'+dx=0$$
where both "x" and "y" are functions of time and {a,b,c,d} are constants, I need a method other than that of Laplace transform, any suggestions?
Edit: Is it possible to solve this system by means of "matrix exponential"?
Is there any textbook on differential equations that covers solving this sort of systems?
Another approach:
Consider the following IVP problem:
\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= -3x - 5y \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= 5x + 3y \end{align}
with $x(0)=x_0$ and $y(0)=y_0$.
Then, Laplace-transform both sides of both equations to get:
\begin{align} s X(s) - x_0 & = - 3 X(s) -5 Y(s) \\ s Y(s) - y_0 & = 5 X(s) + 3Y(s) , \end{align}
which is an algebraic system for $X(s) = \mathcal{L}_sx(t)$ and $Y(s) = \mathcal{L}_sy(t)$. Solve for the unknowns using (for example) Gauss elimination and compute the inverse Laplace transfrom to get the solution.
Cheers!
In case of linear ODE with constant coefficients, it is known that the general solution is a linear combination of exponential functions on the form $k\,e^{r\,t}$ where the various values of $r$ , real or complex, are the roots of a polynomial equation. So, a method to solve the system is to presume of the form of particular solutions and determine the parameters of them, as done below :
Best Answer
The most general approach to these problems is to write your system as a four dimensional first order ODE system: \begin{align} x' &= \xi\\ \xi' &= - b x - a \xi +d y + c \eta\\ y ' &= \eta\\ \eta' &= -d x -c \xi -b y -a \eta \end{align} which can be written in matrix form \begin{equation} \mathbf{x}' = A\, \mathbf{x} \end{equation} with \begin{equation} \mathbf{x} = \begin{pmatrix} x \\ \xi \\ y \\ \eta \end{pmatrix}\qquad \text{and} \qquad A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ -b & -a & d & c \\ 0 & 0 & 0 & 1 \\ -d & -c & -b & -a \end{pmatrix}. \end{equation} Surprisingly, the eigenvalues of this matrix are not particularly hard to find, as are the eigenvectors. The general solution to the above system is then of the form \begin{equation} \mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{x}_1 + c_2 e^{\lambda_2 t}\mathbf{x}_2 + c_3 e^{\lambda_3 t}\mathbf{x}_3 + c_4 e^{\lambda_4 t}\mathbf{x}_4, \end{equation} where $\lambda_{1,2,3,4}$ are the eigenvalues of the matrix $A$ and $\mathbf{x}_{1,2,3,4}$ are the associated eigenvectors. The constants $c_{1,2,3,4}$ are determined by the initial conditions.
As for your question for a literature reference: I can't imagine a textbook on differential equations not treating the above approach, so pick your favourite.