83.19.5 problem 5

Internal problem ID [19162]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (B) at page 55
Problem number : 5
Date solved : Monday, March 31, 2025 at 06:50:42 PM
CAS classification : [_rational, _dAlembert]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 59

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

\[ y = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}} x +\textit {\_Z} +c_1 -{\mathrm e}^{\textit {\_Z}}-x \right )} x +\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +2 \,{\mathrm e}^{\textit {\_Z}} x +\textit {\_Z} +c_1 -{\mathrm e}^{\textit {\_Z}}-x \right )+c_1 -x \]

✓ Mathematica. Time used: 0.845 (sec). Leaf size: 46

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(4*x*y(x) + 1) - 1)/(2*x) cannot be solved by the factorable group method