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Unformatted text preview: Notes for Day Review : . : e Eigenvalue Problem Last time, we discussed the homogeneous system of rst-order equations y = P(t)y, fundamental sets of solutions, the solution matrix, and its Wronskian. Today, we'll discuss nding solutions to rst-order systems in which P(t) is a constant matrix. Introduction
Suppose we had a homogeneous system of rst-order equations y = P(t)y . We remember from our discussion of rst-order equations that the solutions tend to involve functions of the form e t , where is a constant. If we look for a solution to the homogeneous system in this form, say, y= c e t c e t c e t , we get: c e t c e t c e t = P(t) c e t c e t c e t , which is the same thing as saying P(t)y = y. But we recall that this equation is true only when is an eigenvalue of P(t) and y is an eigenvector associated with that eigenvalue. e pair of an eigenvalue and its corresponding eigenvector is called an eigenpair. Today, we'll discuss systems where the eigenpairs are real and distinct; next time, we'll discuss systems with complex eigenpairs. ere is a third case involving repeated eigenvalues that is su ciently complicated that we'll delay treatment of this case until next week. p. , Example Find the general solution of the system: y y = = y + y y +y p. , Example Find the general solution of the system: - - - y = - y In this example, there are repeated eigenvalues. However, we are able to nd distinct eigenvectors, so the overall eigenpairs are distinct. p. , Example Find the general solution of the system: y y y = y -y -y = -y + y - y = -y - y + y e Phase Plane
e phase plane is a type of direction eld (see . ) which we use to visualize the solutions to systems. systems involve three variables--t, y , and y --but in the phase plane, we'll put the variables y and y on the horizontal and vertical axes, and use arrows to indicate the direction of increasing t. Like directions elds, phase planes are tedious to generate by hand, but they're easy to generate using so ware. ese p. , Example
e system y y = = . y + . y . y - . y has a general solution: y(t) = c et + c e -t - . Sketch the phase plane corresponding to this solution. ...
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