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Question Number 108094 by Dwaipayan Shikari last updated on 14/Aug/20

(d^2 Ψ/dt^2 )+(dΨ/dt)+Ψ=0

$$\frac{{d}^{\mathrm{2}} \Psi}{{dt}^{\mathrm{2}} }+\frac{{d}\Psi}{{dt}}+\Psi=\mathrm{0} \\ $$

Answered by mathmax by abdo last updated on 14/Aug/20

y^(′′) +y^′ +y =0 →r^2 +r +1 =0 Δ=1−4 =−3 ⇒r_1 =((−1+i(√3))/2) =e^(i((2π)/3)) and r_2 =((−1−i(√3))/2)=e^(−i((2π)/3)) ⇒y =a e^(r_1 x) +be^(r_2 x) =e^(−(x/2)) {α cos(((√3)/2)x)+βsin(((√3)/2)x)}

$$\mathrm{y}^{''} \:+\mathrm{y}^{'} \:+\mathrm{y}\:=\mathrm{0}\:\rightarrow\mathrm{r}^{\mathrm{2}} +\mathrm{r}\:+\mathrm{1}\:=\mathrm{0} \\ $$$$\Delta=\mathrm{1}−\mathrm{4}\:=−\mathrm{3}\:\Rightarrow\mathrm{r}_{\mathrm{1}} =\frac{−\mathrm{1}+\mathrm{i}\sqrt{\mathrm{3}}}{\mathrm{2}}\:=\mathrm{e}^{\mathrm{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \:\mathrm{and}\:\mathrm{r}_{\mathrm{2}} =\frac{−\mathrm{1}−\mathrm{i}\sqrt{\mathrm{3}}}{\mathrm{2}}=\mathrm{e}^{−\mathrm{i}\frac{\mathrm{2}\pi}{\mathrm{3}}} \\ $$$$\Rightarrow\mathrm{y}\:=\mathrm{a}\:\mathrm{e}^{\mathrm{r}_{\mathrm{1}} \mathrm{x}} \:+\mathrm{be}^{\mathrm{r}_{\mathrm{2}} \mathrm{x}} \:=\mathrm{e}^{−\frac{\mathrm{x}}{\mathrm{2}}} \left\{\alpha\:\mathrm{cos}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{x}\right)+\beta\mathrm{sin}\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{x}\right)\right\} \\ $$

Commented by Dwaipayan Shikari last updated on 14/Aug/20

Thanking you ��

Commented by mathmax by abdo last updated on 14/Aug/20

you are welcome

$$\mathrm{you}\:\mathrm{are}\:\mathrm{welcome} \\ $$

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