University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2023
Views:

227

Pages:

1
Author:

laqala0uk

Maple and Integral Transforms The commands we use are: laplace and invlaplace. Here is an exampleNotice that somewhere in the worksheet (usually at the beginning), we write with(inttrans) First example: Compute some transforms Here, solve the IVP y 00 + y 0 + y = 3 sin(t) y(0) = 1, y 0 (0) = 0 using Laplace transforms and with dsolve. Eqn1:=diff(y(t),t$2)+diff(y(t),t)+y(t)=3*sin(t); Y:=laplace(Eqn1,t,s); Y2:=subs(y(0)=1,D(y)(0)=0,Y); Y3:=solve(Y2,laplace(y(t),t,s)); Y4:=invlaplace(Y3,s,t); dsolve({Eqn1,y(0)=1,D(y)(0)=0},y(t)); Maple and the step function In Maple, the step function is the Heaviside function. That is, Heaviside(t-c)=uc (t) Examples of its use: Solve y 00 + y 0 + y = u2 (t) cos(t − 2) with y(0) = 0 and y 0 (0) = 0. Plot the result. SOLUTION: Eqn1:=diff(y(t),t$2)+diff(y(t),t)+y(t)=Heaviside(t-2)*cos(t-2); Y1:=dsolve({Eqn1,y(0)=0,D(y)(0)=0},y(t)); plot(rhs(Y1),t=0..15); Using the Heaviside function is one way to deal with piecewise defined functions. For example, plot    t if 0 ≤ t ≤ 3 f (t) = 6 − t if 3 < t ≤ 6   0 elsewhere SOLUTION: t(u0 (t) − u3 (t)) + (6 − t)(u3 (t) − u6 (t)) = t(1 − u3 (t)) + (6 − t)(u3 (t) − u6 (t) In Maple: F:=t*(Heaviside(t)-Heaviside(t-3))+(6-t)*(Heaviside(t-3)-Heaviside(t-6)); plot(F,t=-1..7); 1

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