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Mathematics LibreTexts

10.1: Distinct Real Eigenvalues

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We illustrate the solution method by example.

Example: Find the general solution of ˙x1=x1+x2,˙x2=4x1+x2.

View tutorial on YouTube

The equation to be solved may be rewritten in matrix form as

ddt(x1x2)=(1141)(x1x2)

or in short hand as Equation ???.

We take as our ansatz x(t)=veλt, where v and λ are independent of t. Upon substitution into Equation ???, we obtain

λveλt=Avλt

and upon cancellation of the exponential, we obtain the eigenvalue problem

Av=λv.

Finding the characteristic equation using Equation ???, we have

0=det(AλI)=λ22λ3=(λ3)(λ+1)

Therefore, the two eigenvalues are λ1=3 and λ2=1.

To determine the corresponding eigenvectors, we substitute the eigenvalues successively into

(AλI)v=0.

We will write the corresponding eigenvectors v1 and v2 using the matrix notation

(v1v2)=(v11v12v21v22),

where the components of v1 and v2 are written with subscripts corresponding to the first and second columns of a 2-by-2 matrix.

For λ1=3, and unknown eigenvector v1, we have from Equation ???

2v11+v21=04v112v21=0

Clearly, the second equation is just the first equation multiplied by 2, so only one equation is linearly independent. This will always be true, so for the 2 -by-2 case we need only consider the first row of the matrix. The first eigenvector therefore satisfies v21=2v11. Recall that an eigenvector is only unique up to multiplication by a constant: we may therefore take v11=1 for convenience.

For λ2=1, and eigenvector v2=(v12,v22)T, we have from Equation ???

2v12+v22=0,

so that v22=2v12. Here, we take v12=1.

Therefore, our eigenvalues and eigenvectors are given by

λ1=3,v1=(12);λ2=1,v2=(12)

Using the principle of superposition, the general solution to the ode is therefore

X(t)=c1v1eλ1t+c2v2eλ2t

or explicitly writing out the components,

x1(t)=c1e3t+c2etx2(t)=2c1e3t2c2et

We can obtain a new perspective on the solution by drawing a phase portrait, shown in Fig. 10.1, with " x-axis" x1 and " y-axis" x2. Each curve corresponds to a different initial condition, and represents the trajectory of a particle with velocity given by the differential equation. The dark lines represent trajectories along the direction of the eigenvectors. If c2=0, the motion is along the eigenvector v1 with x2=2x1 and the motion with increasing time is away from the origin (arrows pointing out) since the eigenvalue λ1=3>0. If c1=0, the motion is along the eigenvector v2 with x2=2x1 and motion is towards the origin (arrows pointing in) since the eigenvalue λ2=1<0. When the eigenvalues are real and of opposite signs, the origin is called a saddle point. Almost all trajectories (with the exception of those with initial conditions exactly satisfying x2(0)=2x1(0) ) eventually move away from the origin as t increases. When the eigenvalues are real and of the same sign, the origin is called a node. A node can be stable (negative eigenvalues) or unstable (positive eigenvalues).

Screen Shot 2022-05-29 at 10.24.30 PM.png
Figure 10.1: Phase portrait for example with two real eigenvalues of opposite sign.

This page titled 10.1: Distinct Real Eigenvalues is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Jeffrey R. Chasnov via source content that was edited to the style and standards of the LibreTexts platform.

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