2.2.13 Problem 14

Internal problem ID [9136]
Book : Second order enumerated odes
Section : section 2
Problem number : 14
Date solved : Sunday, February 23, 2025 at 05:33:27 AM
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 : 0.008 (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 : 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]
 

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

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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)
 

PolynomialDivisionFailed : couldnt reduce degree in a polynomial division algorithm when dividing [[], [], [], []] by [[ANP([mpq(1,1)], [mpq(1,1), mpq(0,1), mpq(1,1)], QQ)], [ANP([mpq(-1,1), mpq(-1,1)], [mpq(1,1), mpq(0,1), mpq(1,1)], QQ)]]. This can happen when its not possible to detect zero in the coefficient domain. The domain of computation is QQ<I>. Zero detection is guaranteed in this coefficient domain. This may indicate a bug in SymPy or the domain is user defined and doesnt implement zero detection properly.