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Question 21.32: Find the transfer functions of the following equations assum......

Find the transfer functions of the following equations assuming that f(t) represents the input and x(t) represents the output:

\begin{aligned}(a)\ & \frac{\mathrm{d} x(t)}{\mathrm{d} t}-4 x(t)=3 f(t), \quad x(0)=0 \\(b)\ & \frac{\mathrm{d}^2 x(t)}{\mathrm{d} t^2}+3 \frac{\mathrm{d} x(t)}{\mathrm{d} t}-x(t)=f(t), \quad \frac{\mathrm{d} x(0)}{\mathrm{d} t}=0, \quad x(0)=0 \end{aligned}
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(a) Taking Laplace transforms of the differential equation gives

\begin{aligned} & s X(s)-x(0)-4 X(s)=3 F(s) \\ & (s-4) X(s)=3 F(s) \quad \text { as } x(0)=0 \\ & \frac{X(s)}{F(s)}=G(s)=\frac{3}{s-4} \end{aligned}

(b) Taking Laplace transforms of the differential equation gives

\begin{aligned} & s^2 X(s)-s x(0)-\frac{\mathrm{d} x(0)}{\mathrm{d} t}+3(s X(s)-x(0))-X(s)=F(s) \\ & \left(s^2+3 s-1\right) X(s)=F(s) \quad \text { as } \frac{\mathrm{d} x(0)}{\mathrm{d} t}=0 \quad \text { and } \quad x(0)=0 \\ & \frac{X(s)}{F(s)}=G(s)=\frac{1}{s^2+3 s-1} \end{aligned}

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