Analyzing Solution to x'=(({{3, -2}, {4, -1}}))x

This shows one way you can use Mathematica to help check parts of your solution to a system of differential equations. Since x(t) is changing definition as you work through the solution, you cannot go back and evaluate earlier cells once you have worked through the Mathematica file.

Set up the matrix and vector we need:

A={{3,-2},{4,-1}}
MatrixForm[A]

x={x1,x2}
MatrixForm[x]

Plot the vector field

Note I have added some StremPoints to the direction field, which show specific solutions (this is still just a numerical representation of the solution!).

field=VectorPlot[A.x,{x1,-30,30},{x2,-30,30},VectorScale->{Tiny,Small,None},AspectRatio->1,StreamPoints->10,StreamStyle->{Orange,Thick},BaseStyle->20,FrameLabel->{"(Subscript[x, 1])","(Subscript[x, 2])"}]

Solve the System of Differential Equations

Solve for the eigenvalues:

Solve[Det[A-λ*IdentityMatrix[2]]==0,λ]

Solve for the eigenvectors:

Solve[A.x==(1+2 I)x,x]

Notice you need to interpret what you found above, since Mathematica returns the one equation in two unknowns (so pick values of (Subscript[x, 1]) to get values of (Subscript[x, 2])).

Here is a complex valued solution.

sol={1,1-I}Exp[(1+2I)t]
MatrixForm[sol]

You should be able to do this by hand, but this can help you check your result.

sol2=ComplexExpand[sol];
MatrixForm[%]

realpart=Simplify[Re[sol2],Assumptions->t∈Reals]
MatrixForm[realpart]

imaginarypart=Simplify[Im[sol2],Assumptions->t∈Reals]
MatrixForm[imaginarypart]

General Solution:

Clear[x]
x[t_]=c1*realpart+c2*imaginarypart
MatrixForm[%]