2.312   ODE No. 312

1. Problem in Latex
2. Mathematica input
3. Maple input

$$(y(x)y^{\prime }(x)+x)(\frac{x^2}{a}+\frac{y(x{)}^2}{b})+\frac{(a-b)(y(x)y^{\prime }(x)-x)}{a+b}=0$$ ✓ Mathematica : cpu = 0.250683 (sec), leaf count = 204

$$\{\{y(x)\to -\frac{\sqrt{b}\sqrt{a^2+2a^{2}W\left(\frac{c_1(a+b)e^{\frac{b{x}^2}{2a^2}-\frac{b}{2a}-\frac{x^2}{2b}-\frac{1}{2}}}{2a^3{b}^2}\right)+ab-a{x}^2-b{x}^2}}{\sqrt{a}\sqrt{a+b}}\},\{y(x)\to \frac{\sqrt{b}\sqrt{a^2+2a^{2}W\left(\frac{c_1(a+b)e^{\frac{b{x}^2}{2a^2}-\frac{b}{2a}-\frac{x^2}{2b}-\frac{1}{2}}}{2a^3{b}^2}\right)+ab-a{x}^2-b{x}^2}}{\sqrt{a}\sqrt{a+b}}\}\}$$

✓ Maple : cpu = 1.589 (sec), leaf count = 236

$$\{y\left(x\right)=\frac{1}{a}\sqrt{a(-b{x}^2+ab+{\mathrm{e}}^{-\frac{1}{2a^{2}b}(2\mathit{l}\mathit{a}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{t}\mathit{W}\left(1/2\frac{(a+b){\mathrm{e}}^{-1/2}}{a^{2}b}{\mathrm{e}}^{-1/2\frac{x^2}{b}}{\mathrm{e}}^{1/2\frac{b{x}^2}{a^2}}{\mathrm{e}}^{-1/2\frac{b}{a}}{\left({\mathrm{e}}^{\frac{\mathit{\_}\mathit{C}\mathit{1}}{ab}}\right)}^{-1}\right)a^{2}b+a^2{x}^2-b^2{x}^2+a^{2}b+a{b}^2+2\mathit{\_}\mathit{C}\mathit{1}a)})},y\left(x\right)=-\frac{1}{a}\sqrt{a(-b{x}^2+ab+{\mathrm{e}}^{-\frac{1}{2a^{2}b}(2\mathit{l}\mathit{a}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{t}\mathit{W}\left(1/2\frac{(a+b){\mathrm{e}}^{-1/2}}{a^{2}b}{\mathrm{e}}^{-1/2\frac{x^2}{b}}{\mathrm{e}}^{1/2\frac{b{x}^2}{a^2}}{\mathrm{e}}^{-1/2\frac{b}{a}}{\left({\mathrm{e}}^{\frac{\mathit{\_}\mathit{C}\mathit{1}}{ab}}\right)}^{-1}\right)a^{2}b+a^2{x}^2-b^2{x}^2+a^{2}b+a{b}^2+2\mathit{\_}\mathit{C}\mathit{1}a)})}\}$$