Question

please solve this problem

L^-1 { (1/s^2) e ^(-y*sqrt(s+a)) } .

Answer 1

Question #54617, Math / Differential Equations

please solve this problem

$$L^{\wedge} - 1 \left\{ (1/s^{\wedge} 2) e^{\wedge} (-y * \sqrt{s} + t (s + a)) \right\}.$$

Answer.

$$f(t) = L^{-1} \left\{ \frac{1}{s^2} e^{-y\sqrt{s + a}} \right\}$$ $$L^{-1} \left\{ e^{-y\sqrt{s + a}} \right\} = \frac{y e^{-at - \frac{y^2}{4t}}}{2 \pi t^2}$$

As we know $L^{-1} \left\{ \frac{F(s)}{s} \right\} = \int_0^t f(t) dt$

$$\begin{aligned} \text{So} \quad L^{-1} \left\{ \frac{1}{s} e^{-y\sqrt{s + a}} \right\} &= \int_0^t \frac{y e^{-at - \frac{y^2}{4t}}}{2 \pi t^2} dt = \\ &= \frac{1}{2} \left[ e^{-y\sqrt{a}} \operatorname{erfc} \left( \frac{y - 2\sqrt{a}}{2\sqrt{t}} \right) + e^{y\sqrt{a}} \operatorname{erfc} \left( \frac{y + 2\sqrt{a}}{2\sqrt{t}} \right) \right] \\ &= \int_0^t \frac{1}{s^2} e^{-y\sqrt{s + a}} e^{y\sqrt{a}} \operatorname{erfc} \left( \frac{y - 2\sqrt{a}}{2\sqrt{t}} \right) + e^{y\sqrt{a}} \operatorname{erfc} \left( \frac{y + 2\sqrt{a}}{2\sqrt{t}} \right) dt = \\ &= \frac{1}{4a} \left[ (2at - y\sqrt{a}) e^{-y\sqrt{a}} \operatorname{erfc} \left( \frac{y - 2\sqrt{a}}{2\sqrt{t}} \right) + (2at + y\sqrt{a}) e^{y\sqrt{a}} \operatorname{erfc} \left( \frac{y + 2\sqrt{a}}{2\sqrt{t}} \right) \right] \end{aligned}$$

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