Complex Roots Cauchy Euler Equation 5 Equal Roots For F r r r 1 2 0 where r 1

Complex roots cauchy euler equation 5 equal roots for

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Complex Roots Cauchy-Euler Equation 5 Equal Roots : For F ( r ) = ( r - r 1 ) 2 = 0, where r 1 is a double root, then the differential equation L [ y ] = t 2 y 00 + αy 0 + βy = 0 , was shown to satisfy L [ t r 1 ] = 0 and L [ t r 1 ln( t )] = 0 It follows that the general solution is y ( t ) = ( c 1 + c 2 ln( t )) t r 1 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (7/27)
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Cauchy-Euler Equation Review Variation of Parameters Distinct Roots Equal Roots Complex Roots Cauchy-Euler Equation 6 Example: Consider the equation t 2 y 00 + 5 ty 0 + 4 y = 0 By substituting y ( t ) = t r , we have t r [ r ( r - 1) + 5 r + 4] = t r ( r 2 + 4 r + 4) = t r ( r + 2) 2 = 0 This only has the real root r 1 = - 2, which gives general solution y ( t ) = ( c 1 + c 2 ln( t )) t - 2 , t > 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (8/27)
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Cauchy-Euler Equation Review Variation of Parameters Distinct Roots Equal Roots Complex Roots Cauchy-Euler Equation 7 Complex Roots : Assume F ( r ) = 0 has r = μ ± as complex roots, the solutions are still y ( t ) = t r However, t r = e ( μ + ) ln( t ) = t μ [cos( ν ln( t )) + i sin( ν ln( t ))] As before, we obtain the two linearly independent solutions by taking the real and imaginary parts, so the general solution is y ( t ) = t μ [ c 1 cos( ν ln( t )) + c 2 sin( ν ln( t ))] Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (9/27)
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Cauchy-Euler Equation Review Variation of Parameters Distinct Roots Equal Roots Complex Roots Cauchy-Euler Equation 8 Example: Consider the equation t 2 y 00 + ty 0 + y = 0 By substituting y ( t ) = t r , we have t r [ r ( r - 1) + r + 1] = t r ( r 2 + 1) = 0 This has the complex roots r = ± i ( μ = 0 and ν = 1), which gives the general solution y ( t ) = c 1 cos(ln( t )) + c 2 sin(ln( t )) , t > 0 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (10/27)
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Cauchy-Euler Equation Review Variation of Parameters Review Review - Method of undetermined coefficients Applicable for constant coefficient nonhomogeneous linear second order differential equations The nonhomogeneity is limited to sums and products of: Polynomials Exponentials Sines and Cosines Solutions reduce to solving linear equations in the unknown coefficients Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (11/27)
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Cauchy-Euler Equation Review Variation of Parameters Motivating Example Technique of Variation of Parameters Main Theorem for Nonhomogeneous DE Variation of Parameters Variation of Parameters - This method provides a more general method to solve nonhomogeneous problems Technique again begins with a fundamental set of solutions to the homogeneous problem Fundamental set allows creation of the Wronskian Obtain integral formulation from the fundamental solution with the nonhomogeneous function General solution is again formulated from a particular solution added to the homogeneous solution Joseph M. Mahaffy, h [email protected] i Lecture Notes – Second Order Linear Equations — (12/27)
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Cauchy-Euler Equation Review Variation of Parameters Motivating Example
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