2.292   ODE No. 292

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

$$y^{\prime }(x)(ay(x)+bx+c{)}^2+(\alpha y(x)+\beta x+\gamma {)}^2=0$$ ✓ Mathematica : cpu = 62.3396 (sec), leaf count = 760

$$\text{Solve}[(\alpha b-a\beta )\text{RootSum}[{\mathrm{\#}\text{1}}^{3}a{\beta }^3-{\mathrm{\#}\text{1}}^3\alpha b{\beta }^2+2{\mathrm{\#}\text{1}}^{2}a\alpha {\beta }^{2}y(x)+{\mathrm{\#}\text{1}}^{2}a{b}^2\beta y(x)+3\gamma {\mathrm{\#}\text{1}}^{2}a{\beta }^2-2{\mathrm{\#}\text{1}}^2{\alpha }^{2}b\beta y(x)-{\mathrm{\#}\text{1}}^2\alpha b^{3}y(x)-2\gamma {\mathrm{\#}\text{1}}^2\alpha b\beta -{\mathrm{\#}\text{1}}^2\alpha {\beta }^{2}c-\gamma {\mathrm{\#}\text{1}}^2{b}^3+{\mathrm{\#}\text{1}}^2{b}^2\beta c+2\mathrm{\#}\text{1}{a}^{2}b\beta y(x{)}^2+\mathrm{\#}\text{1}a{\alpha }^2\beta y(x{)}^2-2\mathrm{\#}\text{1}a\alpha b^{2}y(x{)}^2+4\gamma \mathrm{\#}\text{1}a\alpha \beta y(x)-2\gamma \mathrm{\#}\text{1}a{b}^{2}y(x)+4\mathrm{\#}\text{1}ab\beta cy(x)+3{\gamma }^2\mathrm{\#}\text{1}a\beta -\mathrm{\#}\text{1}{\alpha }^{3}by(x{)}^2-2\gamma \mathrm{\#}\text{1}{\alpha }^{2}by(x)-2\mathrm{\#}\text{1}{\alpha }^2\beta cy(x)-2\mathrm{\#}\text{1}\alpha b^{2}cy(x)-{\gamma }^2\mathrm{\#}\text{1}\alpha b-2\gamma \mathrm{\#}\text{1}\alpha \beta c-2\gamma \mathrm{\#}\text{1}{b}^{2}c+2\mathrm{\#}\text{1}b\beta c^2+a^3\beta y(x{)}^3-a^2\alpha by(x{)}^3-\gamma a^{2}by(x{)}^2+3a^2\beta cy(x{)}^2+\gamma a{\alpha }^{2}y(x{)}^2-2a\alpha bcy(x{)}^2+2{\gamma }^{2}a\alpha y(x)-2\gamma abcy(x)+3a\beta c^{2}y(x)+{\gamma }^{3}a+{\alpha }^3(-c)y(x{)}^2-2\gamma {\alpha }^{2}cy(x)-\alpha b{c}^{2}y(x)-{\gamma }^2\alpha c-\gamma b{c}^2+\beta c^3\mathrm{\&},\frac{{\mathrm{\#}\text{1}}^2{\beta }^2\log (x-\mathrm{\#}\text{1})+{\alpha }^{2}y(x{)}^2\log (x-\mathrm{\#}\text{1})+2\mathrm{\#}\text{1}\alpha \beta y(x)\log (x-\mathrm{\#}\text{1})+2\gamma \alpha y(x)\log (x-\mathrm{\#}\text{1})+2\gamma \mathrm{\#}\text{1}\beta \log (x-\mathrm{\#}\text{1})+{\gamma }^2\log (x-\mathrm{\#}\text{1})}{-3{\mathrm{\#}\text{1}}^{2}a{\beta }^3+3{\mathrm{\#}\text{1}}^2\alpha b{\beta }^2-4\mathrm{\#}\text{1}a\alpha {\beta }^{2}y(x)-2\mathrm{\#}\text{1}a{b}^2\beta y(x)-6\gamma \mathrm{\#}\text{1}a{\beta }^2+4\mathrm{\#}\text{1}{\alpha }^{2}b\beta y(x)+2\mathrm{\#}\text{1}\alpha b^{3}y(x)+4\gamma \mathrm{\#}\text{1}\alpha b\beta +2\mathrm{\#}\text{1}\alpha {\beta }^{2}c+2\gamma \mathrm{\#}\text{1}{b}^3-2\mathrm{\#}\text{1}{b}^2\beta c-2a^{2}b\beta y(x{)}^2-a{\alpha }^2\beta y(x{)}^2+2a\alpha b^{2}y(x{)}^2-4\gamma a\alpha \beta y(x)+2\gamma a{b}^{2}y(x)-4ab\beta cy(x)-3{\gamma }^{2}a\beta +{\alpha }^{3}by(x{)}^2+2\gamma {\alpha }^{2}by(x)+2{\alpha }^2\beta cy(x)+2\alpha b^{2}cy(x)+{\gamma }^2\alpha b+2\gamma \alpha \beta c+2\gamma b^{2}c-2b\beta c^2}\mathrm{\&}]=c_1,y(x)]$$

✓ Maple : cpu = 0.049 (sec), leaf count = 124

$$\{y\left(x\right)=\frac{1}{-a\beta +b\alpha }(-\gamma b+\beta c+\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\int }^{\mathit{\_}\mathit{Z}}\frac{{\mathit{\_}\mathit{a}}^2{a}^2-2\mathit{\_}\mathit{a}ab+b^2}{{\mathit{\_}\mathit{a}}^3{a}^2-2{\mathit{\_}\mathit{a}}^{2}ab-{\mathit{\_}\mathit{a}}^2{\alpha }^2+2\mathit{\_}\mathit{a}\alpha \beta +\mathit{\_}\mathit{a}{b}^2-{\beta }^2}d\mathit{\_}\mathit{a}+\ln (ax\beta -\alpha bx+a\gamma -\alpha c)+\mathit{\_}\mathit{C}\mathit{1})(x(a\beta -b\alpha )+a\gamma -\alpha c))\}$$