In the following problem apply the eigenvalue method to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. use a computer system or graphingcalculator to construct a direction field and typical solutioncurves for the given system. x'1 = 9x1 + 5x2, x'2 = -6x1 - 2x2 ; x1(0) =1, x2(0)=0
In the following problem apply the eigenvalue method to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. use a computer system or graphingcalculator to construct a direction field and typical solutioncurves for the given system.
x'1 = 9x1 + 5x2, x'2 = -6x1 - 2x2 ; x1(0) =1, x2(0)=0
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We have to find the particular solution of the given system of differential equations. First we have
to find the general solution.
The matrix form of the given system is . Thus, the coefficient matrix is
.
Consider the matrix . The determinant of this matrix is . Solving
the equation , we get, and . These are the eigenvalues.
For , we have, . The null space of this matrix is . It is an eigenvector
corresponding to the eigenvalue 4.
For , we have, . The null space of this matrix is . It is an eigenvector
corresponding to the eigenvalue 3.
Let and .
The general solution is given by where . So we get,
It is given that . So we get,
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