56.1.87 problem 86

Internal problem ID [8799]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 86
Date solved : Wednesday, March 05, 2025 at 06:50:15 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }&=\frac {1}{y}-\frac {x y^{\prime }}{y^{2}} \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 56

ode:=diff(diff(y(x),x),x) = 1/y(x)-x/y(x)^2*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{\operatorname {RootOf}\left (\left (4 \,{\mathrm e}^{\textit {\_Z}} {\cosh \left (\frac {\sqrt {c_{1}^{2}+4}\, \left (2 c_{2} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 c_{1}}\right )}^{2}+c_{1}^{2}+4\right ) x^{2}\right )}-1+\textit {\_Z} c_{1} \right ) x \]

✓ Mathematica. Time used: 0.2 (sec). Leaf size: 77

ode=D[y[x],{x,2}]==1/y[x]-x/y[x]^2*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{2} \log \left (-\frac {y(x)^2}{x^2}-\frac {c_1 y(x)}{x}+1\right )-\frac {c_1 \arctan \left (\frac {\frac {2 y(x)}{x}+c_1}{\sqrt {-4-c_1{}^2}}\right )}{\sqrt {-4-c_1{}^2}}=-\log (x)+c_2,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x)/y(x)**2 + Derivative(y(x), (x, 2)) - 1/y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-y(x)*Derivative(y(x), (x, 2)) + 1)*y(x)/x cannot be solved by the factorable group method