Feb5_ODE

# Feb5_ODE - Response of the 1st Order System to an Arbitrary...

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Response of the 1 st Order System to an Arbitrary Input A 1 st order system is described by an ordinary diﬀerential equation (ODE), a 1 dy ( t ) dt + a 0 y ( t ) = x ( t ) , (1) where a 0 and a 1 are constants. Input x ( t ) is a known function, such that x ( t ) = 0 for t < 0 . (2) We want to ﬁnd the solution of this equation subject to an initial condition (IC), y (0 - ) = 0 . (3) We ﬁrst divide both sides of (1) by a 0 : a 1 a 0 dy ( t ) dt + y ( t ) = 1 a 0 x ( t ) (4) and deﬁne τ = a 1 /a 0 (time constant). Hence, equation to be solved takes form: τ dy ( t ) dt + y ( t ) = 1 a 0 x ( t ) (5) Finding the solution requires two steps: solving a homogeneous equation, and ﬁnding solution to the non-homogeneous equation (5) by variation of parameter. Solution to Homogeneous Equation Homogeneous equation is obtained from (5) by assuming x ( t ) = 0 for all t : τ dy ( t ) dt + y ( t ) = 0 . (6) We re-arrange the terms so that dependent variable y appears only on the left-hand side and independent variable

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Feb5_ODE - Response of the 1st Order System to an Arbitrary...

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