Q5E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Determine, where the periodic function is described by its graph.

Step-by-Step Solution

Answer

The value of\(\mathcal{L}\{ f\} \), where the periodic function is described by its graph is\(\mathcal{L}\{ f(t)\}  = \frac{1}{{s\left( {1 + {e^{ - as}}} \right)}}\).

Step 1: Define the transform of periodic function theorem

If \(f\)has period\(T\)and is piecewise continuous on\([0,T]\) then the Laplace transforms,

are related by,

\({F_T}(s) = F(s)\left[ {1 - {e^{ - sT}}} \right]{\rm{\;or\;}}F(s) = \frac{{{F_T}(s)}}{{1 - {e^{ - sT}}}}\).

Step 2: Use the periodic function from the graph to determine the value

Consider the graph, 

Thus, from the graph the function has the period\(T = 2a\)

Describe the period in the equation,

\({f_T}(t) = \left\{ {\begin{array}{*{20}{l}}{1,0 < t < a}\\{0,a < t < 2a}\end{array}} \right.\)

Hence,

 

Apply the theorem,

\(\begin{array}{c}\mathcal{L}\{ f(t)\}  = \frac{{{F_T}(s)}}{{1 - {e^{ - 2s}}}}\\ = \frac{{1 - {e^{ - as}}}}{{s\left( {1 - {e^{ - 2as}}} \right)}}\\ = \frac{1}{{s\left( {1 + {e^{ - as}}} \right)}}\end{array}\)

Therefore, the value of\(\mathcal{L}\{ f\} \), where the periodic function is described by the given graph is\(\mathcal{L}\{ f(t)\}  = \frac{1}{{s\left( {1 + {e^{ - as}}} \right)}}\).