Transforms of derivatives and ODEs

Differentiation becomes multiplication by s, so initial value problems turn into algebra; unit step functions handle forcing that switches on partway through.

Notes on Diffy Qs: Differential Equations for Engineers, Transforms of derivatives and ODEs

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4 exercises

Exercise 101

Using the Heaviside function $u(t)$ , write down the function

2.5

Exercise 102

Solve $x''-x = (t^2-1) u(t-1)$ for initial conditions $x(0)=1$ , $x'(0) = 2$ using the Laplace transform.

2.5

Exercise 103

Find the transfer function for $x' + x = f(t)$ , $x(0)=0$ .

2.5

Exercise 104

Suppose $Lx = f(t)$ , $x(0)=0$ , $x'(0)=0$ for the input function $f(t) = 4$ gives the output $x(t) = 1 - \cos(2 t)$ . Find $F(s)$ , $X(s)$ , and the transfer function $H(s)$ .

2.5