MHB How to Solve Laplace Transforms with a Fractional Term?

AI Thread Summary
The discussion focuses on solving the inverse Laplace transform of a given expression involving a fractional term. Participants reference standard Laplace transform tables, noting the forms for cosine and sine transforms, and express uncertainty about how to adapt these for a different denominator structure. A participant suggests rewriting the expression to utilize the hyperbolic sine and cosine transforms, leading to a solution involving exponential functions. The conversation concludes with a confirmation of understanding the mathematical approach to proceed with the calculations.
rannasquaer
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How to solve the transforms below

\[ \mathscr{L}^{-1} \frac{a(s+2 \lambda)+b}{(s+ \lambda)^2- \omega^2} \]
 
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rannasquaer said:
How to solve the transforms below

\[ \mathscr{L}^{-1} \frac{a(s+2 \lambda)+b}{(s+ \lambda)^2- \omega^2} \]

The table of Laplace transforms lists that $\mathscr{L}^{-1} \frac{s+\alpha}{(s+\alpha)^2+\omega^2} = e^{-\alpha t}\cos(\omega t)\cdot u(t)$ and $\mathscr{L}^{-1} \frac{\omega}{(s+\alpha)^2+\omega^2} = e^{-\alpha t}\sin(\omega t)\cdot u(t)$.

Can we use those to find the requested transform?
 
Klaas van Aarsen said:
The table of Laplace transforms lists that $\mathscr{L}^{-1} \frac{s+\alpha}{(s+\alpha)^2+\omega^2} = e^{-\alpha t}\cos(\omega t)\cdot u(t)$ and $\mathscr{L}^{-1} \frac{\omega}{(s+\alpha)^2+\omega^2} = e^{-\alpha t}\sin(\omega t)\cdot u(t)$.

Can we use those to find the requested transform?

I think yes, if I rewrite like

\[ \mathscr{L}^{-1} \frac{a(s+\lambda)}{(s+ \lambda)^2- \omega^2} + \mathscr{L}^{-1} \frac{b + \lambda a}{(s+ \lambda)^2- \omega^2} \]

but I have \[ (s+\lambda)^2-\omega^2 \] and not \[ (s+\lambda)^2+\omega^2 \]

The table of Laplace transforms lists that \[ \mathscr{L}^{-1} \frac{\alpha}{s^2- \alpha^2} = \sin h(\alpha t).u(t) \] and \[ \mathscr{L}^{-1} \frac{s}{s^2- \alpha^2} = \cos h(\alpha t).u(t) \]

I do not know what to do now
 
rannasquaer said:
I think yes, if I rewrite like

\[ \mathscr{L}^{-1} \frac{a(s+\lambda)}{(s+ \lambda)^2- \omega^2} + \mathscr{L}^{-1} \frac{b + \lambda a}{(s+ \lambda)^2- \omega^2} \]

but I have \[ (s+\lambda)^2-\omega^2 \] and not \[ (s+\lambda)^2+\omega^2 \]

The table of Laplace transforms lists that \[ \mathscr{L}^{-1} \frac{\alpha}{s^2- \alpha^2} = \sin h(\alpha t).u(t) \] and \[ \mathscr{L}^{-1} \frac{s}{s^2- \alpha^2} = \cos h(\alpha t).u(t) \]

I do not know what to do now

Right. I meant the $\cosh$ and $\sinh$ versions.

Also note that $\mathscr{L}^{-1} F(s-\alpha)=e^{\alpha t}f(t)$.

So we can do:
\[ \mathscr{L}^{-1} \frac{a(s+\lambda)}{(s+ \lambda)^2- \omega^2} + \mathscr{L}^{-1} \frac{b + \lambda a}{(s+ \lambda)^2- \omega^2} =a e^{-\lambda t}\sinh(\omega t) \cdot u(t)+ \frac{b+\lambda a}{\omega}e^{-\lambda t}\cosh(\omega t) \cdot u(t)\]
And if we want to, we can rewrite it using $\sinh x= \frac 12(e^x-e^{-x})$ and $\cosh x=\frac 12(e^x + e^{-x})$.
 
Great, I understood how to continue to do the math. Thank you!😄
 
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