ແກ້ Ya^r+Deltaa-Y=0

$$Y a ^ { r } + \Delta a - Y = 0$$

ແກ້ສຳລັບ Y (complex solution)

$$\left\{\begin{matrix}Y=-\frac{a\Delta }{a^{r}-1}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }a=e^{-\frac{2\pi n_{1}iRe(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}-\frac{2\pi n_{1}Im(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}}\\Y\in \mathrm{C}\text{, }&\Delta =0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }a=e^{-\frac{2\pi n_{1}iRe(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}-\frac{2\pi n_{1}Im(r)}{\left(Re(r)\right)^{2}+\left(Im(r)\right)^{2}}}\end{matrix}\right,$$

ຂັ້ນຕອນສຳລັບການແກ້ສົມຜົນເສັ້ນຊື່

ລົບ \Delta a ອອກຈາກທັງສອງຂ້າງ. ອັນໃດກໍໄດ້ຫານຈາກສູນໄດ້ຈຳນວນລົບຂອງມັນ.

$$Ya^{r}-Y=-\Delta a$$

ຈັດລຳດັບພົດຄືນໃໝ່.

$$Ya^{r}-Y=-a\Delta$$

ຮວມທຸກຄຳສັບທີ່ມີ Y.

$$\left(a^{r}-1\right)Y=-a\Delta$$

ຫານທັງສອງຂ້າງດ້ວຍ a^{r}-1.

$$\frac{\left(a^{r}-1\right)Y}{a^{r}-1}=-\frac{a\Delta }{a^{r}-1}$$

ການຫານດ້ວຍ a^{r}-1 ຈະຍົກເລີກການຄູນດ້ວຍ a^{r}-1.

$$Y=-\frac{a\Delta }{a^{r}-1}$$

ແກ້ສຳລັບ Y

$$\left\{\begin{matrix}Y=-\frac{a\Delta }{a^{r}-1}\text{, }&\left(a=0\text{ and }r>0\right)\text{ or }\left(r\neq 0\text{ and }a\neq -1\text{ and }Denominator(r)\text{bmod}2=1\text{ and }a<0\right)\text{ or }\left(a<0\text{ and }Numerator(r)\text{bmod}2=1\text{ and }Denominator(r)\text{bmod}2=1\right)\text{ or }\left(r\neq 0\text{ and }a\neq 1\text{ and }a>0\right)\\Y\in \mathrm{R}\text{, }&\left(\Delta =0\text{ and }a=-1\text{ and }Numerator(r)\text{bmod}2=0\text{ and }Denominator(r)\text{bmod}2=1\right)\text{ or }\left(\Delta =0\text{ and }a=1\right)\text{ or }\left(\Delta =0\text{ and }a\neq 0\text{ and }r=0\right)\end{matrix}\right,$$

ຂັ້ນຕອນສຳລັບການແກ້ສົມຜົນເສັ້ນຊື່

ລົບ \Delta a ອອກຈາກທັງສອງຂ້າງ. ອັນໃດກໍໄດ້ຫານຈາກສູນໄດ້ຈຳນວນລົບຂອງມັນ.

$$Ya^{r}-Y=-\Delta a$$

ຈັດລຳດັບພົດຄືນໃໝ່.

$$Ya^{r}-Y=-a\Delta$$

ຮວມທຸກຄຳສັບທີ່ມີ Y.

$$\left(a^{r}-1\right)Y=-a\Delta$$

ຫານທັງສອງຂ້າງດ້ວຍ a^{r}-1.

$$\frac{\left(a^{r}-1\right)Y}{a^{r}-1}=-\frac{a\Delta }{a^{r}-1}$$

ການຫານດ້ວຍ a^{r}-1 ຈະຍົກເລີກການຄູນດ້ວຍ a^{r}-1.

$$Y=-\frac{a\Delta }{a^{r}-1}$$