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Nonhomogeneous

# Nonhomogeneous - The Nonhomogeneous Equation Linear...

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The Nonhomogeneous Equation ± Linear Operators and Kernels We have seen that given a second-order linear homogeneous equation y 00 + p ( x ) y 0 + q ( x ) y = 0 we can always deﬁne a linear operator L [ y ] = y 00 + p ( x ) y 0 + q ( x ) y. In this terminology, the function y ( x ) is a solution if and only if L [ y ( x )] = 0 . Given a fundamental set of solutions y 1 and y 2 , we saw that the general solution is y ( x ) = c 1 y 1 + c 2 y 2 This is just a way of writing down the set of all solutions, that is, all functions y ( x ) such that L [ y ( x )] = 0 . Definition Given a linear operator L [ y ] , the kernel of L is the set of all functions y ( x ) such that L [ y ( x )] = 0 . I Example If y 1 and y 2 are a fundamental set of solutions to the homogeneous equation above, the kernel of L is the set of all functions of the form y ( x ) = c 1 y 1 + c 2 y 2 . I Example If L is the linear operator L [ y ] = d dx [ y ] , ﬁnd the kernel of L . Solution Here operator is the derivative operator. The kernel is the set of all functions y ( x ) such that d dx [ y ] = L [ y ( x )] = 0 . But the functions whose derivative is zero are precisely the constant functions. Thus the kernel is the set of all functions of the form y ( x ) = C where C is a constant.

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± The Nonhomogeneous Equation Now suppose we wish to solve the nonhomogeneous equation y 00 + p ( x ) y 0 + q ( x ) y = g ( x ) .
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Nonhomogeneous - The Nonhomogeneous Equation Linear...

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