29.28.13 problem 811

Internal problem ID [5394]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 28
Problem number : 811
Date solved : Tuesday, March 04, 2025 at 09:32:51 PM
CAS classification : [_dAlembert]

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 223

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

✓ Mathematica. Time used: 0.442 (sec). Leaf size: 61

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]
 

\[ \text {Solve}\left [\left \{x=\frac {K[1] \text {arcsinh}(K[1])}{2 \sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=\frac {K[1]}{2}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]

✗ 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