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

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

Solvi għal d

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

Passi biex Issolvi l-Ekwazzjoni Lineari

Immultiplika ż-żewġ naħat tal-ekwazzjoni b'12, l-inqas denominatur komuni ta' 3,2,4.

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

Biex iżżid jew tnaqqas l-espressjonijiet, espandihom biex id-denominaturi tagħhom ikunu l-istess. L-inqas multiplu komuni ta' 3 u 2 huwa 6. Immultiplika \frac{y^{3}}{3} b'\frac{2}{2}. Immultiplika \frac{x^{2}}{2} b'\frac{3}{3}.

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

Billi \frac{2y^{3}}{6} u \frac{3x^{2}}{6} għandhom l-istess denominatur, żidhom billi żżid in-numeraturi tagħhom.

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

Uża l-propjetà distributtiva biex timmultiplika 12 b'y+\frac{2y^{3}+3x^{2}}{6}.

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

Ikkanċella l-akbar fattur komuni 6 f'12 u 6.

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

Uża l-propjetà distributtiva biex timmultiplika 2 b'2y^{3}+3x^{2}.

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

Uża l-propjetà distributtiva biex timmultiplika 12y+4y^{3}+6x^{2} b'd.

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

Uża l-propjetà distributtiva biex timmultiplika 12yd+4y^{3}d+6x^{2}d b'x.

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

Uża l-propjetà distributtiva biex timmultiplika 3 b'x+xy^{2}.

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

Uża l-propjetà distributtiva biex timmultiplika 3x+3xy^{2} b'd.

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

Uża l-propjetà distributtiva biex timmultiplika 3xd+3xy^{2}d b'y.

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

Ikkombina 12ydx u 3xdy biex tikseb 15ydx.

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

Ikkombina 4y^{3}dx u 3xdy^{3} biex tikseb 7y^{3}dx.

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

Ikkombina t-termini kollha li fihom d.

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

L-ekwazzjoni hija f'forma standard.

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

Iddividi 0 b'15yx+7y^{3}x+6x^{3}.

$$d=0$$

Solvi għal x

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