1.54 problem 55

Internal problem ID [2690]

Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number: 55.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_quadrature]

Solve \begin {gather*} \boxed {\left (\sin ^{2}\relax (y)+x \cot \relax (y)\right ) y^{\prime }=0} \end {gather*}

✓ Solution by Maple

Time used: 0.01 (sec). Leaf size: 1223

dsolve((sin(y(x))^2+x*cot(y(x)))*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \arctan \left (-\frac {\sqrt {6 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}-\frac {72 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}}}{6}, \frac {\left (6 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}-\frac {72 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )^{\frac {3}{2}}}{216 x}\right ) \\ y \relax (x ) = \arctan \left (\frac {\sqrt {6 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}-\frac {72 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}}}{6}, -\frac {\left (6 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}-\frac {72 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )^{\frac {3}{2}}}{216 x}\right ) \\ y \relax (x ) = \arctan \left (-\frac {\sqrt {-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}-18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )}}{6}, \frac {\left (-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}-18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}{216 x}\right ) \\ y \relax (x ) = \arctan \left (\frac {\sqrt {-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}-18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )}}{6}, -\frac {\left (-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}-18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}{216 x}\right ) \\ y \relax (x ) = \arctan \left (-\frac {\sqrt {-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}+18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )}}{6}, \frac {\left (-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}+18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}{216 x}\right ) \\ y \relax (x ) = \arctan \left (\frac {\sqrt {-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}+18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )}}{6}, -\frac {\left (-3 \left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}+\frac {36 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}+18 i \sqrt {3}\, \left (\frac {\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2 x^{2}}{\left (108 x^{2}+12 \sqrt {12 x^{6}+81 x^{4}}\right )^{\frac {1}{3}}}\right )\right )^{\frac {3}{2}}}{216 x}\right ) \\ y \relax (x ) = c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.264 (sec). Leaf size: 1367

DSolve[(Sin[y[x]]^2+x*Cot[y[x]])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\text {ArcCos}\left (-\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to \text {ArcCos}\left (-\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to -\text {ArcCos}\left (\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to \text {ArcCos}\left (\sqrt {-\frac {\sqrt [3]{\frac {2}{3}} x^2}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}{\sqrt [3]{2} 3^{2/3}}+1}\right ) \\ y(x)\to -\text {ArcCos}\left (-\sqrt {\frac {x^2 \text {Root}\left [3 \text {$\#$1}^3+2\&,2\right ]}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \text {Root}\left [\text {$\#$1}^6+3888\&,3\right ]-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to \text {ArcCos}\left (-\sqrt {\frac {x^2 \text {Root}\left [3 \text {$\#$1}^3+2\&,2\right ]}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \text {Root}\left [\text {$\#$1}^6+3888\&,3\right ]-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to -\text {ArcCos}\left (\sqrt {\frac {x^2 \text {Root}\left [3 \text {$\#$1}^3+2\&,2\right ]}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \text {Root}\left [\text {$\#$1}^6+3888\&,3\right ]-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to \text {ArcCos}\left (\sqrt {\frac {x^2 \text {Root}\left [3 \text {$\#$1}^3+2\&,2\right ]}{\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (\sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \text {Root}\left [\text {$\#$1}^6+3888\&,3\right ]-2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-27 x^2}+12\right )}\right ) \\ y(x)\to -\text {ArcCos}\left (-\sqrt {1+\frac {1}{12} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \left (\frac {4 \sqrt [3]{-2} x^2}{\left (\sqrt {\frac {4 x^6}{3}+9 x^4}-3 x^2\right )^{2/3}}+\text {Root}\left [\text {$\#$1}^3-96\&,3\right ]\right )}\right ) \\ y(x)\to \text {ArcCos}\left (-\sqrt {1+\frac {1}{12} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \left (\frac {4 \sqrt [3]{-2} x^2}{\left (\sqrt {\frac {4 x^6}{3}+9 x^4}-3 x^2\right )^{2/3}}+\text {Root}\left [\text {$\#$1}^3-96\&,3\right ]\right )}\right ) \\ y(x)\to -\text {ArcCos}\left (\sqrt {1+\frac {1}{12} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \left (\frac {4 \sqrt [3]{-2} x^2}{\left (\sqrt {\frac {4 x^6}{3}+9 x^4}-3 x^2\right )^{2/3}}+\text {Root}\left [\text {$\#$1}^3-96\&,3\right ]\right )}\right ) \\ y(x)\to \text {ArcCos}\left (\sqrt {1+\frac {1}{12} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2} \left (\frac {4 \sqrt [3]{-2} x^2}{\left (\sqrt {\frac {4 x^6}{3}+9 x^4}-3 x^2\right )^{2/3}}+\text {Root}\left [\text {$\#$1}^3-96\&,3\right ]\right )}\right ) \\ y(x)\to c_1 \\ \end{align*}