3.4 - Math 334 Lecture#13 Â 3.4 Reduction of Order Repeated...

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Unformatted text preview: Math 334 Lecture #13 § 3.4: Reduction of Order; Repeated Roots Method of Reduction of Order (d’Alembert). Suppose y 1 is a nonzero solution of y 00 + p ( t ) y + q ( t ) y = 0 . For any scalar c , the scalar multiple cy 1 is also a solution. The idea of d’Alembert to find a solution y 2 such that y 1 , y 2 form a fundamental set of solutions of the ODE is to seek for y 2 in the form of y 2 = v ( t ) y 1 i.e., replace the constant c in the solution cy 1 with a yet to be determined function v ( t ). If there is a nonconstant function v ( t ) for which y 2 = v ( t ) y 1 is a solution of the ODE, then y 1 and y 2 are linearly independent because the ratio y 2 /y 1 = v ( t ) is not a constant. How is a nonconstant v ( t ) found so that y 2 = v ( t ) y 1 is a solution of the ODE? Substitution of the “guess” y 2 = v ( t ) y 1 into the ODE will answer this: since y 2 = v y 1 + vy 1 , and y 00 2 = v 00 + v y 1 + v y 1 + vy 00 1 = v 00 y 1 + 2 v y 1 + vy 00 1 , it follows that 0 = y 00 2 + p ( t ) y 2 + q ( t ) y 2 = v 00 y 1 + 2 v y 1 + vy 00 1 + p ( t )[ v y 1 + vy 1 ] + q ( t ) vy 1 = v 00 y 1 + 2 v y 1 + p ( t ) v y 1 + v y 00 1 + p ( t ) y 1 + q ( t ) y 1 ....
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3.4 - Math 334 Lecture#13 Â 3.4 Reduction of Order Repeated...

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