2.313   ODE No. 313

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

$${\text{Global}\stackrel{`}{}\text{y}}^{\prime }(\text{Global}\stackrel{`}{}\text{x})(3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{x}\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x}{)}^2+2\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x}{)}^3-\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3+\text{Global}\stackrel{`}{}\text{c}{\text{Global}\stackrel{`}{}\text{x}}^2)-\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x}{)}^3+2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3+3\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^2\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x})+\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x}{)}^2=0$$ ✓ Mathematica : cpu = 0.0823304 (sec), leaf count = 537

$$\{\{\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x})\to \frac{\sqrt[3]{2}(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x})}{3\text{Global}\stackrel{`}{}\text{a}\sqrt[3]{\sqrt{(27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3){}^2+4(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x}){}^3}+27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3}}-\frac{\sqrt[3]{\sqrt{(27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3){}^2+4(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x}){}^3}+27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3}}{3\sqrt[3]{2}\text{Global}\stackrel{`}{}\text{a}}\},\{\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x})\to \frac{(1-i\sqrt{3})\sqrt[3]{\sqrt{(27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3){}^2+4(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x}){}^3}+27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3}}{6\sqrt[3]{2}\text{Global}\stackrel{`}{}\text{a}}-\frac{(1+i\sqrt{3})(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x})}{32^{2/3}\text{Global}\stackrel{`}{}\text{a}\sqrt[3]{\sqrt{(27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3){}^2+4(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x}){}^3}+27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3}}\},\{\text{Global}\stackrel{`}{}\text{y}(\text{Global}\stackrel{`}{}\text{x})\to \frac{(1+i\sqrt{3})\sqrt[3]{\sqrt{(27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3){}^2+4(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x}){}^3}+27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3}}{6\sqrt[3]{2}\text{Global}\stackrel{`}{}\text{a}}-\frac{(1-i\sqrt{3})(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x})}{32^{2/3}\text{Global}\stackrel{`}{}\text{a}\sqrt[3]{\sqrt{(27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3){}^2+4(3c_1\text{Global}\stackrel{`}{}\text{a}+3\text{Global}\stackrel{`}{}\text{a}\text{Global}\stackrel{`}{}\text{c}\text{Global}\stackrel{`}{}\text{x}){}^3}+27c_1{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{x}+27{\text{Global}\stackrel{`}{}\text{a}}^2\text{Global}\stackrel{`}{}\text{b}{\text{Global}\stackrel{`}{}\text{x}}^3}}\}\}$$

✓ Maple : cpu = 0.221 (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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\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{-3\frac{-27a{b}^2{x}^6+54\mathit{\_}\mathit{C}\mathit{1}ab{x}^4-4c^3{x}^3-27{\mathit{\_}\mathit{C}\mathit{1}}^{2}a{x}^2+12\mathit{\_}\mathit{C}\mathit{1}{c}^2{x}^2-12{\mathit{\_}\mathit{C}\mathit{1}}^{2}cx+4{\mathit{\_}\mathit{C}\mathit{1}}^3}{a}})a^2}})\}$$