15.20.8 problem 8

Internal problem ID [3316]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 8
Date solved : Sunday, March 30, 2025 at 01:34:47 AM
CAS classification : [_dAlembert]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 304

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

\begin{align*} \frac {\left (-y+\sqrt {y^{2}+4 x}\right ) c_1}{\sqrt {-2 y+2 \sqrt {y^{2}+4 x}+4}\, \sqrt {-2 y+2 \sqrt {y^{2}+4 x}-4}}+x +\frac {\left (-y+\sqrt {y^{2}+4 x}\right ) \left (-\ln \left (2\right )+\ln \left (-y+\sqrt {y^{2}+4 x}+\sqrt {2 y^{2}-2 y \sqrt {y^{2}+4 x}+4 x -4}\right )\right )}{\sqrt {2 y^{2}-2 y \sqrt {y^{2}+4 x}+4 x -4}} &= 0 \\ \frac {\left (y+\sqrt {y^{2}+4 x}\right ) c_1}{\sqrt {-2 y-2 \sqrt {y^{2}+4 x}+4}\, \sqrt {-2 y-2 \sqrt {y^{2}+4 x}-4}}+x -\frac {\left (y+\sqrt {y^{2}+4 x}\right ) \left (-\ln \left (2\right )+\ln \left (-y-\sqrt {y^{2}+4 x}+\sqrt {2 y^{2}+2 y \sqrt {y^{2}+4 x}+4 x -4}\right )\right )}{\sqrt {2 y^{2}+2 y \sqrt {y^{2}+4 x}+4 x -4}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.645 (sec). Leaf size: 59

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x - 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(4*x + y(x)**2)/2 + y(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method