Transformation I

Calculus Level 4

Let $f (t)$ be the inverse Laplace transform of the fraction

$\large \frac {2s} {s^4 + 10s^2 + 9}$ .

If $f( \frac {\pi}{3} )$ can be expressed in the form $\frac {a}{b}$ where $a$ and $b$ are coprime integers, determine $a+b$ .

The answer is 11.

1 solution

Indraneel Mukhopadhyaya
Apr 14, 2017

Firstly note that function in s whose inverse Laplace transform is to be evaluated, is equal to f(s)×g(s) where f(s)=2/(s²+1) and g(s)=s/(s²+9). We know that inverse Laplace transform of f(s) is 2sin(t) and inverse Laplace transform of g(s) is cos(3t). So, applying the Convolution theorem, the answer is integral of 2sin(u)cos(3t-3u) du from u=0 to u=t. The integral comes out to be (1/2)×(sin(t))×(sin(2t)). f (π/3)=3/8. Hence answer is 11

Transformation I

The answer is 11.

1 solution

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