23.2.62 problem 64

Internal problem ID [5417]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 64
Date solved : Tuesday, September 30, 2025 at 12:40:10 PM
CAS classification : [_dAlembert]

\begin{align*} {y^{\prime }}^{2}-2 y y^{\prime }-2 x&=0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 219

ode:=diff(y(x),x)^2-2*y(x)*diff(y(x),x)-2*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {\frac {\left (y-\sqrt {y^{2}+2 x}\right ) \operatorname {arcsinh}\left (-y+\sqrt {y^{2}+2 x}\right )}{2}+x \sqrt {2 y^{2}+2 x -2 y \sqrt {y^{2}+2 x}+1}-2 c_1 y+2 c_1 \sqrt {y^{2}+2 x}}{\sqrt {2 y^{2}+2 x -2 y \sqrt {y^{2}+2 x}+1}} &= 0 \\ \frac {-\frac {\left (y+\sqrt {y^{2}+2 x}\right ) \operatorname {arcsinh}\left (y+\sqrt {y^{2}+2 x}\right )}{2}+x \sqrt {2 y^{2}+2 x +2 y \sqrt {y^{2}+2 x}+1}+2 c_1 y+2 c_1 \sqrt {y^{2}+2 x}}{\sqrt {2 y^{2}+2 x +2 y \sqrt {y^{2}+2 x}+1}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 121

ode=(D[y[x],x])^2-2*y[x]*D[y[x],x]-2*x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=\frac {1}{2} \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ) \int \frac {\exp \left (-\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right )}{K[1]+\frac {1}{K[1]}} \, dK[1]+c_1 \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ),y(x)=\frac {K[1]}{2}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

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