x(x+y)(y)-x^2=K(x^3+y^3) eching

$$x ( x + y ) ( y ) - x ^ { 2 } = K ( x ^ { 3 } + y ^ { 3 } )$$

K uchun yechish

$$\left\{\begin{matrix}K=\frac{x\left(xy-x+y^{2}\right)}{x^{3}+y^{3}}\text{, }&x\neq -y\\K\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right,$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

x ga x+y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(x^{2}+xy\right)y-x^{2}=K\left(x^{3}+y^{3}\right)$$

x^{2}+xy ga y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}y+xy^{2}-x^{2}=K\left(x^{3}+y^{3}\right)$$

K ga x^{3}+y^{3} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}y+xy^{2}-x^{2}=Kx^{3}+Ky^{3}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$Kx^{3}+Ky^{3}=x^{2}y+xy^{2}-x^{2}$$

K'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(x^{3}+y^{3}\right)K=x^{2}y+xy^{2}-x^{2}$$

Tenglama standart shaklda.

$$\left(x^{3}+y^{3}\right)K=yx^{2}+xy^{2}-x^{2}$$

Ikki tarafini x^{3}+y^{3} ga bo‘ling.

$$\frac{\left(x^{3}+y^{3}\right)K}{x^{3}+y^{3}}=\frac{x\left(xy-x+y^{2}\right)}{x^{3}+y^{3}}$$

x^{3}+y^{3} ga bo'lish x^{3}+y^{3} ga ko'paytirishni bekor qiladi.

$$K=\frac{x\left(xy-x+y^{2}\right)}{x^{3}+y^{3}}$$

x\left(-x+y^{2}+yx\right) ni x^{3}+y^{3} ga bo'lish.

$$K=\frac{x\left(xy-x+y^{2}\right)}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}$$