50.1.87 problem 86

Internal problem ID [10085]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 86
Date solved : Tuesday, September 30, 2025 at 06:55:41 PM
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: 0.030 (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);
 

\[ y = \operatorname {RootOf}\left (\textit {\_Z}^{2}-{\mathrm e}^{\operatorname {RootOf}\left (x^{2} \left (4 \,{\mathrm e}^{\textit {\_Z}} {\cosh \left (\frac {\sqrt {c_1^{2}+4}\, \left (2 c_2 +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 c_1}\right )}^{2}+c_1^{2}+4\right )\right )}-1+\textit {\_Z} c_1 \right ) x \]

✓ Mathematica. Time used: 0.149 (sec). Leaf size: 125

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]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {K[3]}{-x^2-c_1 K[3] x+K[3]^2}dK[3]-\int _1^x\left (\frac {\left (c_1+\frac {K[4]}{y(x)}\right ) y(x)}{-K[4]^2-c_1 y(x) K[4]+y(x)^2}+\int _1^{y(x)}-\frac {K[3] (-c_1 K[3]-2 K[4])}{\left (K[3]^2-c_1 K[4] K[3]-K[4]^2\right ){}^2}dK[3]\right )dK[4]=c_2,y(x)\right ] \]

✗ 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