285lect14 - 1 Lecture 14 We observe from the previous...

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1 Lecture 14 We observe from the previous example how one can try to solve the initial value problem y 00 + p ( x ) y 0 + q ( x ) y = 0; y ( x 0 ) = y 0 ; y 0 ( x 0 ) = y 0 0 : We need to know that the particular solution satisfying given initial value problem is unique. y 1 ( x ) ; y 2 ( x ) satisfying the ODE y 00 + p ( x ) y 0 + q ( x ) y = 0 : At this point, we pay no attention to initial conditions. By principle of superposition, y ( x ) = C 1 y 1 ( x ) + C 2 y 2 ( x ) is a solution. Now we try to select constants C 1 and C 2 in such a way that the initial conditions hold for y ( x ) . So, there are two important questions: Is solution of the initial value problem unique? What should we require from y 1 and y 2 to ensure that C 1 and C 2 can be always selected so that the solution y ( x ) = C 1 y 1 ( x ) + C 2 y 2 ( x ) the initial condition? Below, we will discuss the answers to these two fundamental questions. 1.1
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285lect14 - 1 Lecture 14 We observe from the previous...

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