58.2.13 problem 14

Internal problem ID [9136]
Book : Second order enumerated odes
Section : section 2
Problem number : 14
Date solved : Tuesday, January 28, 2025 at 04:00:09 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} 10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 38

dsolve(10*diff(y(x),x$2)+(exp(x)+3*x)*diff(y(x),x)+3/sin(y(x))*exp(y(x))*(diff(y(x),x))^2=0,y(x), singsol=all)
 

\[ \int _{}^{y}{\mathrm e}^{\frac {3 \left (\int \csc \left (\textit {\_b} \right ) {\mathrm e}^{\textit {\_b}}d \textit {\_b} \right )}{10}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\frac {3 x^{2}}{20}-\frac {{\mathrm e}^{x}}{10}}d x \right )-c_{2} = 0 \]

✓ Solution by Mathematica

Time used: 33.212 (sec). Leaf size: 71

DSolve[10*D[y[x],{x,2}]+(Exp[x]+3*x)*D[y[x],x]+3/Sin[y[x]]*Exp[y[x]]*(D[y[x],x])^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (-\int _1^{K[2]}-\frac {3}{10} e^{K[1]} \csc (K[1])dK[1]\right )dK[2]\&\right ]\left [\int _1^x-e^{\frac {1}{20} \left (-3 K[3]^2-2 e^{K[3]}\right )} c_1dK[3]+c_2\right ] \]