(y+y^3/3+x^2/2)dx+1/4(x+xy^2)dy=0 eching

$$( y + \frac { y ^ { 3 } } { 3 } + \frac { x ^ { 2 } } { 2 } ) d x + \frac { 1 } { 4 } ( x + x y ^ { 2 } ) d y = 0$$

d uchun yechish

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }y=\frac{7^{\frac{2}{3}}}{7}\left(\sqrt[3]{\frac{\sqrt{441x^{4}+875}}{7}-3x^{2}}-\sqrt[3]{\frac{\sqrt{441x^{4}+875}}{7}+3x^{2}}\right)\end{matrix}\right,$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

Tenglamaning ikkala tarafini 12 ga, 3,2,4 ning eng kichik karralisiga ko‘paytiring.

$$12\left(y+\frac{y^{3}}{3}+\frac{x^{2}}{2}\right)dx+3\left(x+xy^{2}\right)dy=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3 va 2 ning eng kichik umumiy karralisi 6. \frac{y^{3}}{3} ni \frac{2}{2} marotabaga ko'paytirish. \frac{x^{2}}{2} ni \frac{3}{3} marotabaga ko'paytirish.

$$12\left(y+\frac{2y^{3}}{6}+\frac{3x^{2}}{6}\right)dx+3\left(x+xy^{2}\right)dy=0$$

\frac{2y^{3}}{6} va \frac{3x^{2}}{6} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$12\left(y+\frac{2y^{3}+3x^{2}}{6}\right)dx+3\left(x+xy^{2}\right)dy=0$$

12 ga y+\frac{2y^{3}+3x^{2}}{6} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(12y+12\times \frac{2y^{3}+3x^{2}}{6}\right)dx+3\left(x+xy^{2}\right)dy=0$$

12 va 6 ichida eng katta umumiy 6 faktorini bekor qiling.

$$\left(12y+2\left(2y^{3}+3x^{2}\right)\right)dx+3\left(x+xy^{2}\right)dy=0$$

2 ga 2y^{3}+3x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(12y+4y^{3}+6x^{2}\right)dx+3\left(x+xy^{2}\right)dy=0$$

12y+4y^{3}+6x^{2} ga d ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(12yd+4y^{3}d+6x^{2}d\right)x+3\left(x+xy^{2}\right)dy=0$$

12yd+4y^{3}d+6x^{2}d ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12ydx+4y^{3}dx+6dx^{3}+3\left(x+xy^{2}\right)dy=0$$

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

$$12ydx+4y^{3}dx+6dx^{3}+\left(3x+3xy^{2}\right)dy=0$$

3x+3xy^{2} ga d ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12ydx+4y^{3}dx+6dx^{3}+\left(3xd+3xy^{2}d\right)y=0$$

3xd+3xy^{2}d ga y ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12ydx+4y^{3}dx+6dx^{3}+3xdy+3xdy^{3}=0$$

15ydx ni olish uchun 12ydx va 3xdy ni birlashtirish.

$$15ydx+4y^{3}dx+6dx^{3}+3xdy^{3}=0$$

7y^{3}dx ni olish uchun 4y^{3}dx va 3xdy^{3} ni birlashtirish.

$$15ydx+7y^{3}dx+6dx^{3}=0$$

d'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(15yx+7y^{3}x+6x^{3}\right)d=0$$

Tenglama standart shaklda.

$$\left(6x^{3}+7xy^{3}+15xy\right)d=0$$

0 ni 15yx+7y^{3}x+6x^{3} ga bo'lish.

$$d=0$$

x uchun yechish

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{-42y^{3}-90y}}{6}\text{; }x=-\frac{\sqrt{-42y^{3}-90y}}{6}\text{, }&y\leq 0\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right,$$