Ya^r+Deltaa-Y=0 eching

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

Y uchun yechish (complex solution)

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

Chiziqli tenglamalarni yechish bo'yicha qadamlar

Ikkala tarafdan \Delta a ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

Shartlarni qayta saralash.

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

Y'ga ega bo'lgan barcha shartlarni birlashtirish.

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

Ikki tarafini a^{r}-1 ga bo‘ling.

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

a^{r}-1 ga bo'lish a^{r}-1 ga ko'paytirishni bekor qiladi.

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

Y uchun yechish

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

Chiziqli tenglamalarni yechish bo'yicha qadamlar

Ikkala tarafdan \Delta a ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

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

Shartlarni qayta saralash.

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

Y'ga ega bo'lgan barcha shartlarni birlashtirish.

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

Ikki tarafini a^{r}-1 ga bo‘ling.

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

a^{r}-1 ga bo'lish a^{r}-1 ga ko'paytirishni bekor qiladi.

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