28.1.54 problem 55
Internal
problem
ID
[4360]
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
Date
solved
:
Monday, January 27, 2025 at 09:09:07 AM
CAS
classification
:
[_quadrature]
\begin{align*} \left (\sin \left (y\right )^{2}+x \cot \left (y\right )\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.049 (sec). Leaf size: 1623
dsolve((sin(y(x))^2+x*cot(y(x)))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \arctan \left (-\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right )}{36 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\
y \left (x \right ) &= \arctan \left (\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, -\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right )}{36 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\
y \left (x \right ) &= \arctan \left (-\frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\left (-i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\
y \left (x \right ) &= \arctan \left (\frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right )}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\
y \left (x \right ) &= \arctan \left (-\frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}\, \left (i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right )}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\
y \left (x \right ) &= \arctan \left (\frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{6}, \frac {\left (-i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}-12 x^{2}\right ) \sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{2}/{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}}}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{{1}/{3}}}\right ) \\
y \left (x \right ) &= c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.260 (sec). Leaf size: 1647
DSolve[(Sin[y[x]]^2+x*Cot[y[x]])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\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 \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 -\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 \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 -\arccos \left (-\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 \arccos \left (-\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 -\arccos \left (\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 \arccos \left (\sqrt {\frac {\left (\sqrt {3}-3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (-i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 -\arccos \left (-\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 \arccos \left (-\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 -\arccos \left (\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 \arccos \left (\sqrt {\frac {\left (\sqrt {3}+3 i\right ) x^2}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}}+\frac {1}{12} \left (i 2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {3} \sqrt {x^4 \left (4 x^2+27\right )}-9 x^2}-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 c_1 \\
\end{align*}