2.513   ODE No. 513

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

$$y^{\prime }(x{)}^2\sin (y(x))+2x{y}^{\prime }(x){\cos }^3(y(x))-\sin (y(x)){\cos }^4(y(x))=0$$ ✓ Mathematica : cpu = 0.0790237 (sec), leaf count = 81

$$\{\{y(x)\to {\tan }^{-1}\left(2(-\frac{c_1{}^{3/2}}{\sqrt{x+c_1}}-\frac{\sqrt{c_1}x}{\sqrt{x+c_1}})\right)\},\{y(x)\to {\tan }^{-1}\left(2(\frac{c_1{}^{3/2}}{\sqrt{x+c_1}}+\frac{\sqrt{c_1}x}{\sqrt{x+c_1}})\right)\}\}$$ ✓ Maple : cpu = 4.842 (sec), leaf count = 1134

$$\{[x\left(\mathit{\_}\mathit{T}\right)=\frac{1}{2\mathit{\_}\mathit{T}}({(\cos \left(\frac{1}{2}\arctan (({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2-2\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}}+{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{\frac{2}{3}})\frac{1}{\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}}},-9\frac{1/3{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{2/3}+(\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+1/9\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})(\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}-\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}})}{{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{2/3}})\right))}^4-{\mathit{\_}\mathit{T}}^2)\sin 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(({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2-2\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}}+{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{\frac{2}{3}})\frac{1}{\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}}},-9\frac{1/3{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{2/3}+(\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+1/9\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})(\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}-\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}})}{{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{2/3}})\right){(\cos \left(\frac{1}{2}\arctan (({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2-2\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}}+{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{\frac{2}{3}})\frac{1}{\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}}},-9\frac{1/3{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{2/3}+(\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+1/9\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})(\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}-\sqrt[3]{{\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)}})}{{({\mathit{\_}\mathit{C}\mathit{1}}^3{\mathit{\_}\mathit{T}}^3+54\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1}+6\sqrt{3}\sqrt{{\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2({\mathit{\_}\mathit{C}\mathit{1}}^2{\mathit{\_}\mathit{T}}^2+27)})}^{2/3}})\right))}^{-3},y\left(\mathit{\_}\mathit{T}\right)=\frac{1}{2}\arctan 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