58.2.11 problem 12

Internal problem ID [9134]
Book : Second order enumerated odes
Section : section 2
Problem number : 12
Date solved : Monday, January 27, 2025 at 05:48:38 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 3 y^{\prime \prime }+\cos \left (x \right ) y^{\prime }+\sin \left (y\right ) {y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 27

dsolve(3*diff(y(x),x$2)+cos(x)*diff(y(x),x)+sin(y(x))*(diff(y(x),x))^2=0,y(x), singsol=all)
 

\[ \int _{}^{y}{\mathrm e}^{-\frac {\cos \left (\textit {\_a} \right )}{3}}d \textit {\_a} -c_{1} \left (\int {\mathrm e}^{-\frac {\sin \left (x \right )}{3}}d x \right )-c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 1.388 (sec). Leaf size: 67

DSolve[3*D[y[x],{x,2}]+Cos[x]*D[y[x],x]+Sin[y[x]]*(D[y[x],x])^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[3]}-\frac {1}{3} \sin (K[1])dK[1]\right )dK[3]\&\right ]\left [\int _1^x-\exp \left (-\int _1^{K[4]}\frac {1}{3} \cos (K[2])dK[2]\right ) c_1dK[4]+c_2\right ] \]