2.313   ODE No. 313

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

$$y^{\prime }(x)(3axy(x{)}^2+2ay(x{)}^3-b{x}^3+c{x}^2)-ay(x{)}^3+2b{x}^3+3b{x}^{2}y(x)+cy(x{)}^2=0$$ ✓ Mathematica : cpu = 0.0903104 (sec), leaf count = 537

$$\{\{y(x)\to \frac{\sqrt[3]{2}(3acx+3a{c}_1)}{3a\sqrt[3]{\sqrt{(27a^{2}b{x}^3+27a^2{c}_{1}x){}^2+4(3acx+3a{c}_1){}^3}+27a^{2}b{x}^3+27a^2{c}_{1}x}}-\frac{\sqrt[3]{\sqrt{(27a^{2}b{x}^3+27a^2{c}_{1}x){}^2+4(3acx+3a{c}_1){}^3}+27a^{2}b{x}^3+27a^2{c}_{1}x}}{3\sqrt[3]{2}a}\},\{y(x)\to \frac{(1-i\sqrt{3})\sqrt[3]{\sqrt{(27a^{2}b{x}^3+27a^2{c}_{1}x){}^2+4(3acx+3a{c}_1){}^3}+27a^{2}b{x}^3+27a^2{c}_{1}x}}{6\sqrt[3]{2}a}-\frac{(1+i\sqrt{3})(3acx+3a{c}_1)}{32^{2/3}a\sqrt[3]{\sqrt{(27a^{2}b{x}^3+27a^2{c}_{1}x){}^2+4(3acx+3a{c}_1){}^3}+27a^{2}b{x}^3+27a^2{c}_{1}x}}\},\{y(x)\to \frac{(1+i\sqrt{3})\sqrt[3]{\sqrt{(27a^{2}b{x}^3+27a^2{c}_{1}x){}^2+4(3acx+3a{c}_1){}^3}+27a^{2}b{x}^3+27a^2{c}_{1}x}}{6\sqrt[3]{2}a}-\frac{(1-i\sqrt{3})(3acx+3a{c}_1)}{32^{2/3}a\sqrt[3]{\sqrt{(27a^{2}b{x}^3+27a^2{c}_{1}x){}^2+4(3acx+3a{c}_1){}^3}+27a^{2}b{x}^3+27a^2{c}_{1}x}}\}\}$$

✓ Maple : cpu = 0.218 (sec), leaf count = 912

$$\{y\left(x\right)=\frac{1}{6a}\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}+2(-cx+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}},y\left(x\right)=-\frac{1}{12a}\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}-(-cx+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}}-\frac{i}{2}\sqrt{3}(\frac{1}{6a}\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}-2(-cx+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}}),y\left(x\right)=-\frac{1}{12a}\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}-(-cx+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}}+\frac{i}{2}\sqrt{3}(\frac{1}{6a}\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}-2(-cx+\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(-108b{x}^3+108\mathit{\_}\mathit{C}\mathit{1}x+12\sqrt{-\frac{-81a{b}^2{x}^6+162\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-12c^3{x}^3-81{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+36\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+12{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}})\}$$