Solving Two Dimensional Systems of Linear Differential Equations. 

>	restart; with(LinearAlgebra):

 

A utiity procedure. 

>	buildsys := proc(mat) {diff(x(t),t)=mat[1,1]*x(t)+mat[1,2]*y(t), diff(y(t),t)=mat[2,1]*x(t)+mat[2,2]*y(t)}; end;

 

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Diagonializable, real eigenvalues. 

>	A := Matrix([[-4, 6], [-3, 5]]);

 

>

 

>	sys := buildsys(A);

 

Initial Condition  

>	xno := <a,b>;

 

Maple's Solution 

>	sol := dsolve(sys union {x(0)=a, y(0)=b}, [x(t),y(t)]);

 

Now, we'll solve it. 

>

 

>	evv := Eigenvectors(A);

 

>	P:=evv[2];

 

>	P^(-1).A.P;

 

>	zmat := DiagonalMatrix([exp(2*t), exp(-t)]);

 

>	xsol := P.zmat.P^(-1).xno;

 

Rearrange to compare with Maple. 

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>	map(collect, %, [exp(2*t), exp(-t)]);

 

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Diagonalizable, nonreal eigenvalues. 

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>	A := Matrix([[11, -6], [15, -7]]);

 

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>

 

>	sys := buildsys(A);

 

>	dsol:=dsolve(sys union {x(0)=a, y(0)=b}, [x(t), y(t)]);

 

>

 

>

 

>	evv := Eigenvectors(A);

 

>	P := evv[2];

 

>	p1 := Column(P,1);

 

>	u := map(Re,p1);

 

>	v := -map(Im, p1);

 

>

u - I*v;

 

>	Q := <u|v>;

 

>	B:=Q^(-1).A.Q;

 

>	zmat := << exp(2*t)*cos(3*t)|-exp(2*t)*sin(3*t)>, <exp(2*t)*sin(3*t)| exp(2*t)*cos(3*t)>>;

 

>	xsol := Q.zmat.Q^(-1).xno;

 

To check that this is the same as Maple's solution, create differences 

>	{x(t)-xsol[1], y(t)-xsol[2]};

 

Now substitute Maple's solutions for and  

>	subs(dsol, %);

 

>	map(simplify, %);

 

>

 

Nondiagonalizable. 

>

 

>	A := Matrix([[-8/3, 4/3], [-1/3, -4/3]]);

 

>	sys := buildsys(A);

 

>	dsol := dsolve(sys union {y(0)=b, x(0)=a}, [x(t), y(t)]);

 

>

 

>

 

>	evv := Eigenvectors(A);

 

>	u:= <2, 1>;

 

Pick some vector independent of  

>	v := < -1,3>;

 

>

A.v;

 

>	A.v - (-2)*v;

 

This is Adjust  

>	v := v/(7/3);

 

>	A.v - (-2)*v;

 

This is Tis normalization gives us the 1 in the matrix B below. 

>	P := <u|v>;

 

>	B := P^(-1).A.P;

 

>	zmat := << exp(-2*t) | t*exp(-2*t)>, <0| exp(-2*t)>>;

 

>	xsol := P.zmat.P^(-1).xno;

 

>	{x(t)-xsol[1], y(t)-xsol[2]};

 

>	subs(dsol, %);

 

>	map(simplify, %);

 

>