Notes-2nd order ODE pt2

# Notes-2nd order ODE pt2 - Second Order Linear...

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Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard form y ″ + p ( t ) y ′ + q ( t ) y = g ( t ), g ( t ) ≠ 0. (*) Each such nonhomogeneous equation has a corresponding homogeneous equation : y ″ + p ( t ) y ′ + q ( t ) y = 0. (**) Note that the two equations have the same left-hand side, (**) is just the homogeneous version of (*), with g ( t ) = 0. We will focus our attention to the simpler topic of nonhomogeneous second order linear equations with constant coefficients : a y ″ + b y ′ + c y = g ( t ). Where a , b , and c are constants, a ≠ 0; and g ( t ) ≠ 0. It has a corresponding homogeneous equation a y ″ + b y ′ + c y = 0.

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Solution of the nonhomogeneous linear equations It can be verify easily that the difference y = Y 1 - Y 2 , of any two solutions of the nonhomogeneous equation (*), is always a solution of its corresponding homogeneous equation (**). Therefore, every solution of (*) can be obtained from a single solution of (*), by adding to it all possible solutions of its corresponding homogeneous equation (**). As a result: Theroem : The general solution of the second order nonhomogeneous linear equation y ″ + p ( t ) y ′ + q ( t ) y = g ( t ) can be expressed in the form y = y c + Y where Y is any specific function that satisfies the nonhomogeneous equation, and y c = C 1 y 1 + C 2 y 2 is a general solution of the corresponding homogeneous equation y ″ + p ( t ) y ′ + q ( t ) y = 0 . (That is, y 1 and y 2 are a pair of fundamental solutions of the corresponding homogeneous equation; C 1 and C 2 are arbitrary constants.) The term y c = C 1 y 1 + C 2 y 2 is called the complementary solution (or the homogeneous solution ) of the nonhomogeneous equation. The term Y is called the particular solution (or the nonhomogeneous solution ) of the same equation.
In the case of nonhomgeneous equations with constant coefficients, the complementary solution can be easily found from the roots of the characteristic polynomial. They are always one of the three forms: t r t r c e C e C y 2 1 2 1 + = y c = C 1 e λ t cos μ t + C 2 e λ t sin μ t y c = C 1 e rt + C 2 t e rt Therefore, the only task remaining is to find the particular solution Y , which is any one function that satisfies the given nonhomogeneous equation. That might sound like an easy task. But it is quite nontrivial. There are two general approaches to find Y : the Methods of Undetermined Coefficients , and Variation of Parameters . We will only study the former in this class.

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Method of Undetermined Coefficients The Method of Undetermined Coefficients (sometimes referred to as the method of Judicious Guessing ) is a systematic way (almost, but not quite, like using “educated guesses”) to determine the general form/type of the particular solution Y ( t ) based on the nonhomogeneous term g ( t ) in the given
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Notes-2nd order ODE pt2 - Second Order Linear...

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