Transcription of 16 Laplace transform. Solving linear ODE - NDSU
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16 laplace transform . Solving linear ODEI this lecture I will explain how to use the Laplace transform to solve an ODE with constant main tool we will need is the following property from the last lecture:5 (t)g=F(s). ThenL{f (t)}=sF(s) f(0);L{f (t)}=s2F(s) sf(0) f (0):Now consider a second order IVPy +py +qy=f(t);y(0) =y0;y (0) =y1:(1)I consider a second order equation here, but it should be clear that similar considerations will lead toa solution of any order linear differential equation with constant the Laplace transform to the left and right hand sides of ODE (1):L{y +py +qy}=Lffg=)L{y }+pL{y }+qLfyg=Lffg=)s2 Lfyg sy(0) y (0) +psLfyg py(0) =Lffg=)(s2+ps+q)Lfyg sy0 y1 py0=Lffg=)(s2+ps+q)Y(s) sy0 y1 py0=F(s) =)Y(s) =F(s) +sy0+y1+py0s2+ps+q;where I used the notationY(s) =LfygandF(s) =Lffg.
16 Laplace transform. Solving linear ODE ... I consider a second order equation here, but it should be clear that similar considerations will lead to a solution of any order linear differential equation with constant coefficients.
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Second Order Differential Equations, Chapter 2 Second Order Differential Equations, Order Linear Ordinary Differential Equations, Equations, Order, Second, Order differential, NUMERICALSOLUTIONOF ORDINARYDIFFERENTIAL, NUMERICALSOLUTIONOF ORDINARYDIFFERENTIAL EQUATIONS, Order differential equations, DIFFERENTIAL EQUATIONS, Reduction of Order, Order Equations, Differential, Special Second Order Equations Sect, Special Second order, Second order, Second order differential, For Linear Systems of Differential Equations, Second order equations{Undetermined, Applications of Di erential Equations