Are the entries rational functions? How many parameters? Are you looking for a particular eigenvalue?

Try LinearAlgebra:-CharacteristicPolynomial.

Try setting one parameter to 0 or 1.

@mvcr15 Wikipedia has good information. Let's start with Euler's method for a single equation. That's the easiest case. You should be able to find a discussion of it in any serious calculus textbook. Or see the Wikipedia article "Euler method".

@abbeykabir Okay, good. Yes, the Jacobian matrix is the derivative of a multivariate function. The single-variable Newton's method divides by the derivative. In the multivariate case the derivative is a matrix. How can we divide by a matrix?

@abbeykabir Okay, good. Yes, the Jacobian matrix is the derivative of a multivariate function. The single-variable Newton's method divides by the derivative. In the multivariate case the derivative is a matrix. How can we divide by a matrix?

Do you need the methods to solve a single first-order differential equation, or do you need it for a system of equations?

@abbeykabir Ok, that's good. Now let's generalize that to the multivariate case, where x is an n-vector, and f is a vector-valued function on n-vectors (i.e., f: R^n -> R^n). In particluar, what corresponds to the derivative in the multivariable case?

@abbeykabir Ok, that's good. Now let's generalize that to the multivariate case, where x is an n-vector, and f is a vector-valued function on n-vectors (i.e., f: R^n -> R^n). In particluar, what corresponds to the derivative in the multivariable case?

laplace suffers from a severe case of "floating-point contagion"---the phenomenon that an expression with some floats acquires more floats after a transformation. You can see what's going on from lines 10-15 of showstat(inttrans[laplace]): If the original has any floats, then all rationals (including integers) in the result are converted to floats. The cure is as ThU says: Apply convert(..., rational) before calling laplace. Of course, as ThU said, laplace is not going to be of much help in solving this differential equation.

laplace suffers from a severe case of "floating-point contagion"---the phenomenon that an expression with some floats acquires more floats after a transformation. You can see what's going on from lines 10-15 of showstat(inttrans[laplace]): If the original has any floats, then all rationals (including integers) in the result are converted to floats. The cure is as ThU says: Apply convert(..., rational) before calling laplace. Of course, as ThU said, laplace is not going to be of much help in solving this differential equation.

@abbeykabir Let's start with what you do know. What do you know about Newton's method? Let's start with the single-variable case.

@abbeykabir Let's start with what you do know. What do you know about Newton's method? Let's start with the single-variable case.

I ran your code exactly as it is above in Maple 17.01 and I got that II, the symbolic imaginary part, is identically 0.

@abbeykabir Did he say that the answers are numerically wrong? Or did he just want you to use another method? There are likely to be multiple solutions to the system. Do you have any information about acceptable ranges for the y's? or some method to test whether a solution is feasible?

I'll write up an example using Newton's method.

@abbeykabir Did he say that the answers are numerically wrong? Or did he just want you to use another method? There are likely to be multiple solutions to the system. Do you have any information about acceptable ranges for the y's? or some method to test whether a solution is feasible?

I'll write up an example using Newton's method.

@puckie Yes, sorry, that statement is quite complicated. But note that it got significantly more complicated when you cut and pasted it. Learning that statement will take quite a bit of study. A place to start is ?LinearAlgebra,General,MVshortcut and ?$