Thus the general solution is given by x 1 t x 2 t c 1 2 1 c 2 2 1 e 4 t Joseph

Thus the general solution is given by x 1 t x 2 t c 1

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Thus the general solution is given by x 1 ( t ) x 2 ( t ) = c 1 2 1 + c 2 2 - 1 e - 4 t . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (27/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Real and Different Eigenvalues 14 Example 4 (cont): The phase portrait for x 1 ( t ) x 2 ( t ) = c 1 2 1 + c 2 2 - 1 e - 4 t . This is a degenerate case where the line x 1 = 2 x 2 all form equilibria . All solutions exponentially approach one of the equilibria along lines parallel to the line x 1 = - 2 x 2 Note: There is an unstable case , which we omit, where the eigenvalues satisfy λ 1 = 0 and λ 2 > 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (28/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Complex Eigenvalues 1 Consider a system of two linear homogeneous differential equations : ˙ x = Ax , where A is a real-valued matrix. With a solution of the form x ( t ) = v e λt , there are eigenvalues , λ , with corresponding eigenvectors , v satisfying det | A - λ I | = 0 and ( A - λ I ) v = 0 The characteristic equation for the eigenvalues is a quadratic equation. Assume the eigenvalues are complex, then λ = μ ± , since A is real-valued Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (29/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Complex Eigenvalues 2 Assume the DE, ˙ x = Ax , has eigenvalues λ 1 = μ + and λ 2 = ¯ λ 1 = μ - Assume v 1 is an eigenvector corresponding to λ 1 , so ( A - λ 1 I ) v 1 = 0 Taking conjugates (with A , I , and 0 , real) ( A - ¯ λ 1 I v 1 = ( A - λ 2 I v 1 = 0 This gives two complex solutions to the system of DEs x 1 ( t ) = e ( μ + ) t v 1 and x 2 ( t ) = e ( μ - ) t ¯ v 1 We use Euler’s formula to separate the solutions into real and imaginary parts e iνt = cos( νt ) + i sin( νt ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Systems of Two First Order Eq — (30/54)
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Introduction Solutions of Two 1 st Order Linear DEs Homogeneous Linear System of Autonomous DEs Case Studies and Bifurcation Real and Different Eigenvalues Complex Eigenvalues Repeated Eigenvalues Bifurcation Example and Stability Diagram Complex Eigenvalues 3 Assume the eigenvector , v 1 = a + i b , where a and b are real-valued, then x 1 ( t ) = ( a + i b ) e μt (cos( νt ) + i sin( νt )) = e μt ( a cos( νt ) - b sin( νt )) + ie μt ( a sin( νt ) + b cos( νt )) Denote the real and imaginary parts of x 1 ( t ) = u ( t ) + i w ( t ) u ( t ) = e μt ( a cos( νt ) - b sin( νt )) and w ( t ) = e μt ( a sin( νt )+ b cos( νt )) A similar calculation gives x 2 ( t ) = u ( t ) - i w ( t ) , so x 1 ( t ) and x 2 ( t ) are complex conjugates.
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