Discussion
The inverse Laplace transform of \(\frac{2s^{2}}{s^{4} + 5s^{2} + 4}\) is :
Answer: A
The inverse transform is \(\frac{4}{3}\sin(2x) - \frac{2}{3}\sin(x)\), \(x \gt 0\).
The denominator factors as \((s^{2}+1)(s^{2}+4)\). Write the fraction in partial fractions:
\[ \begin{aligned} \frac{2s^{2}}{(s^{2}+1)(s^{2}+4)} &= \frac{A}{s^{2}+1} + \frac{B}{s^{2}+4} \\ 2s^{2} &= A(s^{2}+4) + B(s^{2}+1) \end{aligned} \]
Putting \(s^{2} = -1\) gives \(-2 = 3A\), so \(A = -2/3\); putting \(s^{2} = -4\) gives \(-8 = -3B\), so \(B = 8/3\). Using \(\mathcal{L}^{-1}[1/(s^{2}+a^{2})] = \sin(ax)/a\):
\[ \begin{aligned} f(x) &= -\frac{2}{3}\sin x + \frac{8}{3}\cdot\frac{\sin 2x}{2} \\ &= \frac{4}{3}\sin 2x - \frac{2}{3}\sin x \end{aligned} \]
Check at small \(x\): the transform behaves like \(2/s^{2}\) for large \(s\), so \(f(x) \approx 2x\); indeed \(\frac{8}{3}x - \frac{2}{3}x = 2x\). The other sign and coefficient combinations fail this check.
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