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EGR 1101: Unit 12 Lecture #1 Differential Equations (Sections 10.1 to 10.4 of Rattan/Klingbeil text) Linear ODE with Constant Coefficients Given independent variable t and dependent variable y(t), a linear ordinary differential equation with constant coefficients is an equation of the form n An d y dt n ... A1 dy dt A0 y ( t ) f ( t ) where A0, A1, …, An, are constants. Some Examples Examples of linear ordinary differential equation with constant coefficients: dy 2 y 8 dt 2 d y dt 2 5 d y dt 6 y 3t dt 4 3 dy 4 3 d y dt 3 8 dy dt sin t Forcing Function In the equation n An d y dt n ... A1 dy dt A0 y ( t ) f ( t ) the function f(t) is called the forcing function. It can be a constant (including 0) or a function of t, but it cannot be a function of y. Solving Linear ODEs with Constant Coefficients • Solving one of these equations means finding a function y(t) that satisfies the equation. You already know how to solve some of these equations, such as dy 2 dt • But many equations are more complicated and cannot be solved just by integrating. A Procedure for Solving Linear ODEs with Constant Coefficients We’ll use a four-step procedure for solving this type of equation: 1. 2. 3. 4. Find the transient solution. Find the steady-state solution. Find the total solution by adding the results of Steps 1 and 2. Apply initial conditions (if given) to evaluate unknown constants that arose in the previous steps. See pages 371-372 in Rattan/Klingbeil textbook. Forcing Function = 0? If the forcing function (the right-hand side of your differential equation) is equal to 0, then the steady-state solution is also 0. In such cases, you get to skip straight from Step 1 to Step 3! Some Equations that Our Procedure Can’t Handle Nonlinear differential equations 2 dy 3 2 ( 7 y ) sin t dt Partial differential equations 2 y t 7 y x t 2 Diff eqs whose coefficients depend on y or t 2t dy dt 7y 0 MATLAB Commands Without initial conditions: >>dsolve('2*Dy + y = 8') With initial conditions: >>dsolve('2*Dy + y = 8', 'y(0)=5') MATLAB Commands Without initial conditions: >>dsolve('D2y+5*Dy+6*y=3*t') With initial conditions: >>dsolve('D2y+5*Dy+6*y=3*t', 'y(0)=0', 'Dy(0)=0') Today’s Examples 1. 2. Leaking bucket with constant inflow rate and bucket initially empty Leaking bucket with zero inflow and bucket initially filled to a given level EGR 1101: Unit 12 Lecture #2 First-Order Differential Equations in Electrical Systems (Section 10.4 of Rattan/Klingbeil text) Review: Procedure Steps in solving a linear ordinary differential equation with constant coefficients: 1. 2. 3. 4. Find the transient solution. Find the steady-state solution. Find the total solution by adding the results of Steps 1 and 2. Apply initial conditions (if given) to evaluate unknown constants that arose in the previous steps. Forcing Function = 0? Recall that if the forcing function (the righthand side of your differential equation) is equal to 0, then the steady-state solution is also 0. In such cases, you get to skip straight from Step 1 to Step 3. Today’s Examples 1. 2. Series RC circuit with constant source voltage First-order low-pass filter Exponentially Saturating Function A function of the form = (1 − −/ ) where K and are constants, is called an exponentially saturating function. At t = 0, f(t) = 0. As t , f(t) K. Exponentially Saturating Function: Time Constant In = (1 − −/ ), the quantity is called the time constant. The time constant is a measure of how quickly or slowly the function rises. The greater is, the more slowly the function approaches its limiting value K. Time Constant Rules of Thumb For = (1 − −/ ), When t = , f(t) 0.632 K. (After one time constant, the function has risen to about 63.2% of its limiting value.) When t = 5 , f(t) 0.993 K. (After five time constants, the function has risen to about 99.3% of its limiting value.) See next slide for graph. Exponentially Saturating Function: Graph Exponentially Decaying Function A function of the form = −/ where K and are constants, is called an exponentially decaying function. At t = 0, f(t) = K. As t , f(t) 0. Exponentially Decaying Function: Time Constant In = −/ , the quantity is called the time constant. The time constant is a measure of how quickly or slowly the function falls. The greater is, the more slowly the function approaches 0. Time Constant Rules of Thumb For = −/ , When t = , f(t) 0.368 K. (After one time constant, the function has fallen to about 36.8% of its initial value.) When t = 5 , f(t) 0.007 K. (After five time constants, the function has fallen to about 0.7% of its initial value.) See next slide for graph. Exponentially Decaying Function: Graph Low-Pass and High-Pass Filters A low-pass filter is a circuit that passes low-frequency signals and blocks highfrequency signals. A high-pass filter is a circuit that does just the opposite: it blocks low-frequency signals and passes high-frequency signals.