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.
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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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