2.2.13 problem 14

Internal problem ID [8548]
Book : Second order enumerated odes
Section : section 2
Problem number : 14
Date solved : Sunday, November 10, 2024 at 09:12:30 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

\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*}

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful`
 

Maple dsolve solution

Solving time : 0.010 (sec)
Leaf size : 38

dsolve(10*diff(diff(y(x),x),x)+(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)
 

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

Mathematica DSolve solution

Solving time : 0.265 (sec)
Leaf size : 90

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]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\exp \left (\left (-\frac {3}{10}-\frac {3 i}{10}\right ) e^{(1+i) K[1]} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i K[1]}\right )\right )dK[1]\&\right ]\left [\int _1^x-e^{\frac {1}{20} \left (-3 K[2]^2-2 e^{K[2]}\right )} c_1dK[2]+c_2\right ] \]