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

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

Whakaoti mō d

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

Ngā Upane mō te Whakaoti Whārite Rārangi

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2,4.

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

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 3 me 2 ko 6. Whakareatia \frac{y^{3}}{3} ki te \frac{2}{2}. Whakareatia \frac{x^{2}}{2} ki te \frac{3}{3}.

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

Tā te mea he rite te tauraro o \frac{2y^{3}}{6} me \frac{3x^{2}}{6}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te y+\frac{2y^{3}+3x^{2}}{6}.

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

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te 12 me te 6.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 2y^{3}+3x^{2}.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12y+4y^{3}+6x^{2} ki te d.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12yd+4y^{3}d+6x^{2}d ki te x.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+xy^{2}.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+3xy^{2} ki te d.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3xd+3xy^{2}d ki te y.

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

Pahekotia te 12ydx me 3xdy, ka 15ydx.

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

Pahekotia te 4y^{3}dx me 3xdy^{3}, ka 7y^{3}dx.

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

Pahekotia ngā kīanga tau katoa e whai ana i te d.

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

He hanga arowhānui tō te whārite.

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

Whakawehe 0 ki te 15yx+7y^{3}x+6x^{3}.

$$d=0$$

Whakaoti mō x

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