How to get Maple to do such an example

Consider the planar system with coefficient matrix

>	B:=matrix(2,2,[[2,1],[-9,2]]);

We can then find the eigenvectors and eigenvalues using eigenvectors.

>	evects(B):=eigenvectors(B);

We can then define the complex solution using the first eigenvalue and eigenvector.

>	xcomplex:=proc(t) exp([evects(B)][1][1]*t)*[evects(B)][1][3][1][1] end: ycomplex:=proc(t) exp([evects(B)][1][1]*t)*[evects(B)][1][3][1][2] end: xcomplex(t); ycomplex(t);

We get real solutions by taking real and imaginary parts.

>	xreal:=proc(t) evalc(Re(xcomplex(t))) end: yreal:=proc(t) evalc(Re(ycomplex(t))) end: xim:=proc(t) evalc(Im(xcomplex(t))) end: yim:=proc(t) evalc(Im(ycomplex(t))) end: matrix(2,1,[[xreal(t)],[yreal(t)]]); matrix(2,1,[[xim(t)],[yim(t)]]);

And here's what the phase plane looks like.

>	phaseportrait([D(x)(t)=2*x(t)+y(t),D(y)(t)=-9*x(t)+2*y(t)], [x(t),y(t)],t=0..2*Pi/3, [[x(0)=1/3,y(0)=1],[x(0)=-1/3,y(0)=1], [x(0)=1/3,y(0)=-1],[x(0)=-1/3,y(0)=-1]], linecolor=[blue,navy,green,cyan], stepsize=0.01);

>	plot([xreal(t),yreal(t)],t=0..4*Pi/3);

A Linear Second Order Equation Example

This one we have seen before.

Consider the equation

.

It has exponential solutions for numbers satisfying the associated quadratic equation

,

that is, and .  Indeed,  the associated first order system is

,

,

and the coefficient matrix is

>	A:=matrix([[0,1],[-50,-2]]);

with characteristic polynomial

>	charpoly(A,s);

and eigenvalues

>	eigenvalues(A);

and the eigenvectors are

>	evalA:=eigenvectors(A);

Corresponding to the first eigenvalue and eigenvector we have the solutions

>	ycomplex:=proc(t) exp((-1+7*I)*t) end; vcomplex:=proc(t) (-1+7*I)*exp((-1+7*I)*t) end;

We get two independent solutions by taking real and imaginary parts.

>	yreal:=proc(t) evalc(Re(ycomplex(t))) end; vreal:=proc(t) evalc(Re(vcomplex(t))) end; yim:=proc(t) evalc(Im(ycomplex(t))) end; vim:=proc(t) evalc(Im(vcomplex(t))) end;

And these give us the solutions we obtained by the guess and test method.

>	matrix(2,1,[[yreal(t)],[vreal(t)]]); matrix(2,1,[[yim(t)],[vim(t)]]);

Indeed, we found one solution to be

>	matrix(2,1,[[exp(-t)*sin(7*t)],[diff(exp(-t)*sin(7*t),t)]]);

The phase portrait we have seen before.

>	phaseportrait([D(y)(t)=v(t),D(v)(t)=-50*y(t)-2*v(t)], [y(t),v(t)],t=0..2, [[y(0)=1,v(0)=0],[y(0)=-1,v(0)=0]], linecolor=[navy,green], stepsize=0.01);

And here again are the y and v functions plotted.

>	plot([exp(-t)*sin(7*t),diff(exp(-t)*sin(7*t),t)],t=0..4, title="y in red, v in green");