Internal problem ID [3199]
Internal file name [OUTPUT/2691_Sunday_June_05_2022_08_38_51_AM_59993157/index.tex]
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.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {\left (\sin \left (y\right )^{2}+x \cot \left (y\right )\right ) y^{\prime }=0} \] The ode \begin {align*} \left (\sin \left (y\right )^{2}+x \cot \left (y\right )\right ) y^{\prime } = 0 \end {align*}
Gives the following equations \begin {align*} \sin \left (y\right )^{2}+x \cot \left (y\right ) = 0\tag {1} \\ y^{\prime } = 0\tag {2} \\ \end {align*}
Each of the above equations is now solved.
Solving ODE (1) Since \(\sin \left (y\right )^{2}+x \cot \left (y\right ) = 0\), is missing derivative in \(y\) then it is an algebraic equation. Solving for \(y\). \begin {align*} \end {align*}
Solving ODE (2) Integrating both sides gives \begin {align*} y &= \int { 0\,\mathop {\mathrm {d}x}}\\ &= c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \] Verified OK.
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \\ \end{align*}
Verification of solutions
\[ y = c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\sin \left (y\right )^{2}+x \cot \left (y\right )\right ) y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=0 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 0d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 1635
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 )^{\frac {2}{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{6}, \frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}\, \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}-12 x^{2}\right )}{36 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{6}, -\frac {\sqrt {\frac {\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}-12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}\, \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}-12 x^{2}\right )}{36 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{6}, \frac {\left (-i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{6}, \frac {\sqrt {\frac {i \left (-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}-12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}\, \left (i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}-12 x^{2}\right )}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{6}, \frac {\sqrt {\frac {i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}\, \left (i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}\right )}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{6}, \frac {\left (-i \left (\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}+\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {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 )^{\frac {2}{3}}+12 x^{2}\right ) \sqrt {3}-\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {2}{3}}+12 x^{2}}{\left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}}}{72 x \left (108 x^{2}+12 \sqrt {3}\, \sqrt {4 x^{6}+27 x^{4}}\right )^{\frac {1}{3}}}\right ) \\ y \left (x \right ) &= c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 1647
DSolve[(Sin[y[x]]^2+x*Cot[y[x]])*y'[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*}