4.1 - Math 334 Lecture#14 4.1 General Theory of n...

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Unformatted text preview: Math 334 Lecture #14 § 4.1: General Theory of n th-order Linear ODE’s An n th-order linear ODE has the form L [ y ] = d n y dt n + p 1 ( t ) d n- 1 y dt n- 1 + · · · + p n- 1 ( t ) dy dt + p n ( t ) y = g ( t ) . [The linear differential operator L acts on n-times differentiable functions.] Specifying n initial conditions of the form y ( t ) = y , y ( t ) = y , . . . y ( n- 1) ( t ) = y ( n- 1) , determines an initial value problem. Existence and Uniqueness Theorem. If the functions p 1 , p 2 , . . . , p n and g are continuous on an open interval I containing t , then the IVP has a unique solution y = φ ( t ) which is n-times differentiable on I . The Homogeneous Linear ODE. Suppose functions y 1 , y 2 , . . . , y n are solutions of L [ y ] = 0. Then the arbitrary linear combination, c 1 y 1 + c 2 y 2 + · · · + c n y n , is also a solution of L [ y ] = 0 by the linearity of L . Are all of the solutions of L [ y ] = 0 given by such a linear combination?...
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4.1 - Math 334 Lecture#14 4.1 General Theory of n...

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