69.8.16 problem 214

Internal problem ID [18119]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8. First order not solved for the derivative. Exercises page 67
Problem number : 214
Date solved : Thursday, October 02, 2025 at 02:43:20 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 71

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

\begin{align*} y &= \frac {4 c_1 \operatorname {LambertW}\left (-\frac {\sqrt {x}}{2}\right )^{2}+2 x \operatorname {LambertW}\left (-\frac {\sqrt {x}}{2}\right )+x}{4 \operatorname {LambertW}\left (-\frac {\sqrt {x}}{2}\right )^{2}} \\ y &= \frac {4 c_1 \operatorname {LambertW}\left (\frac {\sqrt {x}}{2}\right )^{2}+2 x \operatorname {LambertW}\left (\frac {\sqrt {x}}{2}\right )+x}{4 \operatorname {LambertW}\left (\frac {\sqrt {x}}{2}\right )^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 67

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

\begin{align*} y(x)&\to \int _1^x\frac {1}{2 W\left (-\frac {1}{2 \sqrt {\frac {1}{K[1]}}}\right )}dK[1]+c_1\\ y(x)&\to \int _1^x\frac {1}{2 W\left (\frac {1}{2 \sqrt {\frac {1}{K[2]}}}\right )}dK[2]+c_1 \end{align*}

✗ Sympy

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

NotImplementedError : multiple generators [_X0, exp(1/_X0)] 
No algorithms are implemented to solve equation _X0**2*x - exp(1/_X0)