2.404   ODE No. 404

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

$$a{y}^{\prime }(x{)}^2+b{x}^2{y}^{\prime }(x)+cxy(x)=0$$ ✓ Mathematica : cpu = 2.40845 (sec), leaf count = 795

$$\{\text{Solve}[{\int }_1^x(\frac{(3b+2c)K[1{]}^2}{2(3bK[1{]}^3+cK[1{]}^3+9ay(x))}+\frac{3\sqrt{K[1](b^{2}K[1{]}^3-4acy(x))}}{2(3bK[1{]}^3+cK[1{]}^3+9ay(x))})dK[1]+{\int }_1^{y(x)}(\frac{9\sqrt{x(b^2{x}^3-4acK[2])}a}{2(3b+c)x^2(3b{x}^3+c{x}^3+9aK[2])}+\frac{3(3b+2c)a}{2(3b+c)(3b{x}^3+c{x}^3+9aK[2])}-{\int }_1^x(-\frac{9a(3b+2c)K[1{]}^2}{2{(3bK[1{]}^3+cK[1{]}^3+9aK[2])}^2}-\frac{3acK[1]}{(3bK[1{]}^3+cK[1{]}^3+9aK[2])\sqrt{K[1](b^{2}K[1{]}^3-4acK[2])}}-\frac{27a\sqrt{K[1](b^{2}K[1{]}^3-4acK[2])}}{2{(3bK[1{]}^3+cK[1{]}^3+9aK[2])}^2})dK[1]-\frac{\sqrt{x(b^2{x}^3-4acK[2])}}{2(3b+c)x^{2}K[2]}+\frac{b}{2(3b+c)K[2]})dK[2]=c_1,y(x)],\text{Solve}[{\int }_1^x(\frac{(3b+2c)K[3{]}^2}{2(3bK[3{]}^3+cK[3{]}^3+9ay(x))}-\frac{3\sqrt{K[3](b^{2}K[3{]}^3-4acy(x))}}{2(3bK[3{]}^3+cK[3{]}^3+9ay(x))})dK[3]+{\int }_1^{y(x)}(-\frac{9\sqrt{x(b^2{x}^3-4acK[4])}a}{2(3b+c)x^2(3b{x}^3+c{x}^3+9aK[4])}+\frac{3(3b+2c)a}{2(3b+c)(3b{x}^3+c{x}^3+9aK[4])}-{\int }_1^x(-\frac{9a(3b+2c)K[3{]}^2}{2{(3bK[3{]}^3+cK[3{]}^3+9aK[4])}^2}+\frac{3acK[3]}{(3bK[3{]}^3+cK[3{]}^3+9aK[4])\sqrt{K[3](b^{2}K[3{]}^3-4acK[4])}}+\frac{27a\sqrt{K[3](b^{2}K[3{]}^3-4acK[4])}}{2{(3bK[3{]}^3+cK[3{]}^3+9aK[4])}^2})dK[3]+\frac{\sqrt{x(b^2{x}^3-4acK[4])}}{2(3b+c)x^{2}K[4]}+\frac{b}{2(3b+c)K[4]})dK[4]=c_1,y(x)]\}$$ ✓ Maple : cpu = 0.527 (sec), leaf count = 389

$$\{{\int }_{\mathit{\_}\mathit{b}}^x(-b{\mathit{\_}\mathit{a}}^2-\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}acy\left(x\right)}){(b{\mathit{\_}\mathit{a}}^3+\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}acy\left(x\right)}\mathit{\_}\mathit{a}+6ay\left(x\right))}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}-2\frac{a}{b{x}^3+\sqrt{b^2{x}^4-4\mathit{\_}\mathit{f}acx}x+6a\mathit{\_}\mathit{f}}-{\int }_{\mathit{\_}\mathit{b}}^{x}6\frac{a}{{(b{\mathit{\_}\mathit{a}}^3+\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac}\mathit{\_}\mathit{a}+6a\mathit{\_}\mathit{f})}^2}(2\frac{\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac}{\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac}}+b{\mathit{\_}\mathit{a}}^2+\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac})\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0,{\int }_{\mathit{\_}\mathit{b}}^x(-b{\mathit{\_}\mathit{a}}^2+\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}acy\left(x\right)}){(b{\mathit{\_}\mathit{a}}^3+6ay\left(x\right)-\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}acy\left(x\right)}\mathit{\_}\mathit{a})}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}2\frac{a}{-b{x}^3+\sqrt{b^2{x}^4-4\mathit{\_}\mathit{f}acx}x-6a\mathit{\_}\mathit{f}}-{\int }_{\mathit{\_}\mathit{b}}^x-6\frac{a}{{(b{\mathit{\_}\mathit{a}}^3+6a\mathit{\_}\mathit{f}-\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac}\mathit{\_}\mathit{a})}^2}(2\frac{\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac}{\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac}}-b{\mathit{\_}\mathit{a}}^2+\sqrt{{\mathit{\_}\mathit{a}}^4{b}^2-4\mathit{\_}\mathit{a}\mathit{\_}\mathit{f}ac})\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$