Laplace Transforms with Non-repeated Irreducible quadratic factors

Question

I'm solving the following differential equation with a Laplace Transform:

$$x″+ 9x = \cos(t) + \delta(t-\pi)$$

The initial conditions are that x(0) and x'(0) are equal to 0.

After after the transform, we obtain:

$$F(S) = \frac{S}{(S^2+9)[(S^2+1)+9]} + \frac{e^{-\pi\cdot t}}{S^2+9}$$

The problem is that there are two unique irreducible quadratic factors in the denominator of that first term: therefore I cannot seem to find a way to simplify this to find equivalent components which would make the inverse transform easier. How do I find the inverse of that first term?

$$\frac{S}{(S^2+9)[(S^2+1)+9]}$$

The answer includes simple cos(t) and cos(3t) terms with coefficients of $\frac{1}{8}$ and $\frac{-1}{8}$ respectively. What is the process of going from this complex laplace transform to something so simple?

Answer 1

Alternative Method

$$ F(s)=\underbrace{\frac{s}{(s^2+1)(s^2+9)}}_{\mathbf{(I)}}+\underbrace{\frac{e^{-\pi s}}{s^2+9}}_{\mathbf{(II)}} $$

Alternative (and fast) method for $\mathbf{(I)}$:

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We can also decompose $\mathbf{(I)}$ using partial fractions by the following way:

$$ \frac{s}{(s^2+1)(s^2+9)}\equiv\frac{Es+F}{s^2+1}+\frac{Gs+H}{s^2+9} $$ Finding the constants of partial fractions decomposition $$ \frac{s}{(s^2+1)(s^2+9)}\equiv\frac{(Es+F)\cdot(s^2+9)+(Gs+H)\cdot(s^2+1)}{(s^2+1)(s^2+9)} $$

Now, working only with the numerators: $$s\equiv (E+G)\cdot s^3+(F+H)\cdot s^2+(9E+G)\cdot s+(9F+H)$$ Thus, we can find $E$, $F$, $G$ and $H$ solving the following system of equations:

$$\mbox{$ \begin{cases} \begin{alignat}{1} E+G&=0 \\F+H&=0 \\9E+G&=1 \\9F+H&=0 \end{alignat} \end{cases} $} \implies \mbox{$ \begin{cases} \begin{alignat}{1} E&=1/8 \\F&=0 \\G&=-1/8 \\H&=0 \end{alignat} \end{cases} $} $$

Thus, we get: $$ \frac{s}{(s^2+1)(s^2+9)}\equiv \frac{s}{8\cdot(s^2+1)}-\frac{s}{8\cdot(s^2+9)} $$

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Now we can apply the Inverse Laplace Transform:

Remember that the Laplace transform of cosine function is: $$ \cos(\omega t) \iff \frac{s}{s^2+\omega^2} $$ So, we have:

$$ \begin{alignat}{1} \mathscr{L}^{-1}\left[\frac{s}{8\cdot(s^2+1)}-\frac{s}{8\cdot(s^2+9)}\right]&=\frac 1 8 \,\mathscr{L}^{-1}\left[\frac{s}{s^2+1^2}\right]-\frac 1 8 \,\mathscr{L}^{-1}\left[\frac{s}{s^2+3^2}\right] \\&=\frac{1}{8}\left[cos(t)-cos(3t)\right] \end{alignat} $$

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Same answer here

$$ \bbox[5px,border:1.1px solid black] { x(t)=\frac{1}{8}\left[cos(t)-cos(3t)\right]-\frac 1 3\sin(3t)\cdot\operatorname{H}(t-\pi) } $$

Answer 2

Laplace Transform

We have, as pointed in the question, the following equation: $$x''(t)+ 9x(t) = \cos(t) + \delta(t-\pi)$$ With $x(0)=0$ and $x'(0)=0$. So applying the Laplace Transform: $$ \require{cancel} \begin{alignat}{1} \mathscr{L}\left[x''(t)+9x(t)\right]&=\mathscr{L}\left[ \cos(t) + \delta(t-\pi)\right] \\ s^2F(s)-s\cancelto{0}{f(0)}-\cancelto{0}{f'(0)}+9F(s)&=\frac{s}{s^2+1^2}+e^{-\pi s}\cdot1 \\s^2F(s)+9F(s)&=\frac{s}{s^2+1}+e^{-\pi s} \\(s^2+9)\cdot F(s)&=\frac{s+(s^2+1) e^{-\pi s}}{s^2+1} \end{alignat}$$

So, after the transform, we obtain: $$ \bbox[5px,border:1.1px solid black] { F(s)=\frac{s+(s^2+1) e^{-\pi s}}{(s^2+1)(s^2+9)}=\frac{s}{(s^2+1)(s^2+9)}+\frac{e^{-\pi s}}{s^2+9} } $$

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Your transform is wrong in two places: $$F(S) = \frac{S}{(S^2+9)[(S^2+1) \bbox[yellow,4px,border:1.5px solid red]{+9}\,]} + \frac{e^{-\pi\cdot \, \bbox[yellow,4px,border:1.5px solid red]t}}{S^2+9}$$

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Inverse Laplace Transform

Now, we have two parts in the right side to apply the Inverse Laplace Transform: $$ F(s)=\underbrace{\frac{s}{(s^2+1)(s^2+9)}}_{\mathbf{(I)}}+\underbrace{\frac{e^{-\pi s}}{s^2+9}}_{\mathbf{(II)}} $$

For $\mathbf{(I)}$, we need to decompose it via partial fractions first:

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$\,\,\,\,\,\,\,\mathbf{1.}$ Finding zeros of $s^2+1$ and $s^2+9$ $$\begin{alignat}{1} s^2+1=0 &\iff s=\pm\sqrt{-1}=\pm \,i \\s^2+9=0 &\iff s=\pm\sqrt{-3}=\pm \,3i \end{alignat} $$ Now that we know that $s^2+1=(s+i)\cdot(s-i)$ and $s^2+9=(s+3i)\cdot(s-3i)$, we can proceed the decomposition.

$\,\,\,\,\,\,\,\mathbf{2.}$ Finding the constants of partial fractions decomposition $$ \frac{s}{(s^2+1)(s^2+9)}\equiv\frac{A}{s+i}+\frac{B}{s-i}+\frac{C}{s+3i}+\frac{D}{s-3i}\equiv \\\equiv\frac{A\cdot (s-i)(s^2+9)+B\cdot(s+i)(s^2+9)+C\cdot (s^2+1)(s-3i)+D\cdot (s^2+1)(s+3i)}{(s^2+1)(s^2+9)} $$ Now, working only with the numerators: $$s\equiv A\cdot (s-i)(s^2+9)+B\cdot(s+i)(s^2+9)+C\cdot (s^2+1)(s-3i)+D\cdot (s^2+1)(s+3i)$$ Since we're dealing with an identity (i.e. valid for every $s$), we can arbitrarily chose values and put them into the identity to find $A$, $B$, $C$ and $D$:

$$ \begin{cases} \begin{alignat}{1} s=-i &\implies A=\frac{1}{16} \\ s=i &\implies B=\frac{1}{16} \\ s=-3i &\implies C=-\frac{1}{16} \\ s=3i &\implies D=-\frac{1}{16} \end{alignat} \end{cases} $$ Thus, we get: $$ \frac{s}{(s^2+1)(s^2+9)}\equiv\frac{1/16}{s+i}+\frac{1/16}{s-i}-\frac{1/16}{s+3i}-\frac{1/16}{s-3i} $$

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Now we can apply the Inverse Laplace Transform: $$ \mathscr{L}^{-1}\left[\frac{1/16}{s+i}+\frac{1/16}{s-i}-\frac{1/16}{s+3i}-\frac{1/16}{s-3i}\right]= \\=\frac{1}{16}\,\mathscr{L}^{-1}\left[\frac{1}{s+i}\right]+\frac{1}{16}\,\mathscr{L}^{-1}\left[\frac{1}{s-i}\right]-\frac{1}{16}\,\mathscr{L}^{-1}\left[\frac{1}{s+3i}\right]-\frac{1}{16}\,\mathscr{L}^{-1}\left[\frac{1}{s-3i}\right] $$ Remembering that the Laplace transform of exponential function is: $$ e^{-at} \iff \frac{1}{s+a} $$ We finally have: $$ \mathscr{L}^{-1}\left[\frac{1/16}{s+i}+\frac{1/16}{s-i}-\frac{1/16}{s+3i}-\frac{1/16}{s-3i}\right]= \frac{1}{16}e^{-i\cdot t}+\frac{1}{16}e^{i\cdot t}-\frac{1}{16}e^{-3i\cdot t}-\frac{1}{16}e^{3i\cdot t} $$ Using Euler's identity ($e^{i\cdot\theta}\equiv \cos\theta+i\sin\theta)$, we can write the above result in terms of sines and cosines: $$ \begin{cases} \begin{alignat}{1} e^{-i\cdot t}&=\cos(t)-i\sin(t) \\e^{i\cdot t}&=\cos(t)+i\sin(t) \\e^{-3i\cdot t}&=\cos(3t)-i\sin(3t) \\e^{3i\cdot t}&=\cos(3t)+i\sin(3t) \end{alignat} \end{cases} \implies \frac{1}{16}\left(e^{-i\cdot t}+e^{i\cdot t}-e^{-3i\cdot t}-e^{3i\cdot t}\right)=\frac{1}{8}\left[cos(t)-cos(3t)\right] $$

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For $\mathbf{(II)}$, we need to use the following:

$\,\,\,\,\,\,\,\mathbf{1.}$ Laplace time shift property $$ f(t-a) \cdot \operatorname{H}(t-a) \iff e^{-a s}\cdot F(s) $$ (where $\operatorname{H}(t)$ is the Heaviside step function)

$\,\,\,\,\,\,\,\mathbf{2.}$ Laplace transform of sine function $$ \sin(\omega t) \iff \frac{\omega}{s^2+\omega^2} $$

Combining these two things, we can find the Inverse Laplace Transform of $\mathbf{(II)}$:

$$ \mathscr{L}^{-1}\left[\frac{e^{-\pi s}}{s^2+9}\right]=\mathscr{L}^{-1}\left[\frac{1} {\color{red}3}\cdot\frac{\color{red}3\cdot e^{-\pi s}}{s^2+3^2}\right]=\frac 1 3 \cdot \mathscr{L}^{-1}\left[e^{-\pi s}\cdot\frac{3}{s^2+3^2}\right]=\frac 1 3\sin(3t-\pi)\cdot\operatorname{H}(t-\pi) $$

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Finally, the answer:

Remembering that $\sin(3t-\pi)=-\sin(3t)$, we have: $$ \bbox[5px,border:1.1px solid black] { x(t)=\frac{1}{8}\left[cos(t)-cos(3t)\right]-\frac 1 3\sin(3t)\cdot\operatorname{H}(t-\pi) } $$