Internal problem ID [7454]
Internal file name [OUTPUT/6421_Sunday_June_05_2022_04_51_50_PM_70711030/index.tex]
Book: Second order enumerated odes
Section: section 2
Problem number: 14.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {10 y^{\prime \prime }+\left ({\mathrm e}^{x}+3 x \right ) y^{\prime }+\frac {3 \,{\mathrm e}^{y} {y^{\prime }}^{2}}{\sin \left (y\right )}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 0.0 (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)
\[ \int _{}^{y \left (x \right )}{\mathrm e}^{\frac {3 \left (\int \csc \left (\textit {\_b} \right ) {\mathrm e}^{\textit {\_b}}d \textit {\_b} \right )}{10}}d \textit {\_b} -c_{1} \left (\int {\mathrm e}^{-\frac {3 x^{2}}{20}-\frac {{\mathrm e}^{x}}{10}}d x \right )-c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.234 (sec). Leaf size: 90
DSolve[10*y''[x]+(Exp[x]+3*x)*y'[x]+3/Sin[y[x]]*Exp[y[x]]*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (\left (-\frac {3}{10}-\frac {3 i}{10}\right ) e^{(1+i) K[1]} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i K[1]}\right )\right )dK[1]\&\right ]\left [\int _1^x-e^{\frac {1}{20} \left (-3 K[2]^2-2 e^{K[2]}\right )} c_1dK[2]+c_2\right ] \]