Question

Laplace transform of t cos (at) is
1. \(\frac{{{s^2} + {a^2}}}{{{{\left( {{s^2} - {a^2}} \right)}^2}}}\)
2. \(\frac{s}{{{{\left( {{s^2} - {a^2}} \right)}^2}}}\)
3. \(\frac{{{s^2} - {a^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)
4. \(\frac{s}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)

Answer 1

Correct Answer - Option 3 : \(\frac{{{s^2} - {a^2}}}{{{{\left( {{s^2} + {a^2}} \right)}^2}}}\)

Concept:

Laplace transform of f(t):

L {f(t)} = F (s)

Then, Multiplication by tⁿ:           

L {tⁿ f(t)} = (-1)^{n\(\frac{{{d^n}}}{{d{s^n}}}\)} [ F(s)]

Now,

L (cos at) = \(\frac{{\rm{s}}}{{{{\rm{s}}^2} + {{\rm{a}}^2}}}\)

L {t cos at} = (-1) \(\frac{{\rm{d}}}{{{\rm{ds}}}}\left( {\frac{{\rm{s}}}{{{{\rm{s}}^2} + {{\rm{a}}^2}}}} \right)\)

= \(\frac{{{\rm{s\;}}\left( {2{\rm{s}}} \right) - \left( 1 \right)\left( {{{\rm{s}}^2} + {{\rm{a}}^2}} \right)}}{{{{\left( {{{\rm{s}}^2} + {{\rm{a}}^2}} \right)}^2}}}\)

= \(\frac{{{{\rm{s}}^2} - {{\rm{a}}^2}}}{{{{\left( {{{\rm{s}}^2} + {{\rm{a}}^2}} \right)}^2}}}\)