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UnkleRhaukus
 2 years ago
Evaluate \[\int\limits_0^x\text{erf}\left(\sqrt{xt}\right)\text{erf}\left(\sqrt t\right)\text dt\]
UnkleRhaukus
 2 years ago
Evaluate \[\int\limits_0^x\text{erf}\left(\sqrt{xt}\right)\text{erf}\left(\sqrt t\right)\text dt\]

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UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0\[\begin{align*} y(t)&=\int\limits_0^x\text{erf}\left(\sqrt{xt}\right)\text{erf}\left(\sqrt t\right)\text dt\\ Y(p)&=\mathcal L\left\{\int\limits_0^x\text{erf}\left(\sqrt{xt}\right)\text{erf}\left(\sqrt t\right)\text dt\right\}\\ \\&=\mathcal L\left\{\text{erf}\left(\sqrt t\right)\ast\text{erf}\left(\sqrt t\right)\right\}\\ \\&=\frac {1}{p\sqrt{p+1}}\times\frac {1}{p\sqrt{p+1}}\\ \\&=\frac {1}{p^2(p+1)}\\ \end{align*}\]

UnkleRhaukus
 2 years ago
Best ResponseYou've already chosen the best response.0is this right/?, how can i check?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1This appears to be correct to me, so far. :) I checked it here just by making sure each step was justified and matched up correctly to the definitions... line 12: taking laplace transform of both sides \(\checkmark\) line 23: definition of convolution. \(\checkmark\) line 34: convolution theorem \(\checkmark\) etc.
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