62.17.10 problem Ex 10

Internal problem ID [12839]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 28. Summary. Page 59
Problem number : Ex 10
Date solved : Monday, March 31, 2025 at 07:21:08 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.305 (sec). Leaf size: 87

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

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

✓ Mathematica. Time used: 1.13 (sec). Leaf size: 178

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

\begin{align*} \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{x \sqrt {x^4+4 y(x)^2}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\frac {i x^2}{2} \\ y(x)\to \frac {i x^2}{2} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x*(-x**2 + sqrt(x**4 + 4*y(x)**2))/(2*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method