2.2.13 Problem 14

Internal problem ID [10424]
Book : Second order enumerated odes
Section : section 2
Problem number : 14
Date solved : Monday, December 08, 2025 at 08:50:34 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*}

2.2.13.1 ✓ Maple. Time used: 0.004 (sec). Leaf size: 38

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ \int _{}^{y}{\mathrm e}^{\frac {3 \int {\mathrm e}^{\textit {\_b}} \csc \left (\textit {\_b} \right )d \textit {\_b}}{10}}d \textit {\_b} -c_1 \int {\mathrm e}^{-\frac {3 x^{2}}{20}-\frac {{\mathrm e}^{x}}{10}}d x -c_2 = 0 \]

Maple trace

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

2.2.13.2 ✓ Mathematica. Time used: 32.727 (sec). Leaf size: 71

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ] \end{align*}

2.2.13.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x + exp(x))*Derivative(y(x), x) + 3*exp(y(x))*Derivative(y(x), x)**2/sin(y(x)) + 10*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out