Q1Advanced Engineering Mathematics

The mathematical problem strictly requires the evaluation of the inverse Laplace transform for the rational function $F(s)=\frac{2s^2-1}{(s^2+1)(s^2+4)}$. Because the denominator consists of non-repeating, irreducible quadratic factors, we cannot rely on immediate table lookups. The optimal algebraic strategy is to decompose this complex fraction into simpler, manageable partial fractions.

1. Algebraic Decomposition Strategy

To significantly simplify the intense algebraic manipulations, we employ a temporary substitution. We let $x=s^2$. This elegantly reduces our function to $G(x)=\frac{2x-1}{(x+1)(x+4)}$. This transformed expression can now be decomposed into standard linear partial fractions:

We eliminate the denominators by multiplying the entire equation by the common multiple $(x+1)(x+4)$. This yields the foundational identity equation:

We now strategically solve for the unknown coefficients $A$ and $B$ by substituting specific values for $x$ that intentionally eliminate terms. First, we substitute $x=-1$. This eliminates the $B$ term: $2(-1)-1=A(-1+4)⟹-3=3A⟹A=-1$. Next, we substitute $x=-4$. This eliminates the $A$ term: $2(-4)-1=B(-4+1)⟹-9=-3B⟹B=3$.

2. Applying the Inverse Transform

Having successfully determined the exact coefficients, we revert our temporary substitution ($x=s^2$) to reconstruct the original frequency-domain Laplace function:

Because the inverse Laplace transform operator $L^{-1}$ is strictly linear, we can independently evaluate the transform of each isolated term. We reference the standard, foundational time-domain transformation pair $L^{-1}\left\{\frac{a}{s^2+a^2}\right\}=\sin (at)$.

For the first fraction, $a=1$, yielding $-\sin (t)$. For the second fraction, the denominator is $s^2+2^2$, meaning $a=2$. To match the standard formula exactly, we must multiply and divide the numerator by 2, resulting in $\frac{3}{2}\left(\frac{2}{s^2+2^2}\right)$, which transforms seamlessly into $\frac{3}{2}\sin (2t)$.

Combining these two independently derived components yields the final, exact time-domain analytical solution: $f(t)=\frac{3}{2}\sin (2t)-\sin (t)$.