50.1.23 problem 24

Internal problem ID [10021]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 24
Date solved : Tuesday, September 30, 2025 at 06:46:54 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.596 (sec). Leaf size: 97

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

\begin{align*} \ln \left (x \right )-\sqrt {1+4 y}-\frac {\ln \left (\sqrt {1+4 y}-1\right )}{2}+\frac {\ln \left (1+\sqrt {1+4 y}\right )}{2}-\frac {\ln \left (y\right )}{2}-c_1 &= 0 \\ \ln \left (x \right )+\sqrt {1+4 y}+\frac {\ln \left (\sqrt {1+4 y}-1\right )}{2}-\frac {\ln \left (1+\sqrt {1+4 y}\right )}{2}-\frac {\ln \left (y\right )}{2}-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 12.805 (sec). Leaf size: 72

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

\begin{align*} y(x)&\to \frac {1}{4} W\left (-e^{-1-2 c_1} x\right ) \left (2+W\left (-e^{-1-2 c_1} x\right )\right )\\ y(x)&\to \frac {1}{4} W\left (e^{-1+2 c_1} x\right ) \left (2+W\left (e^{-1+2 c_1} x\right )\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.448 (sec). Leaf size: 65

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

\[ \left [ \frac {\sqrt {4 y{\left (x \right )} + 1}}{2} + \frac {\log {\left (x \right )}}{2} - \frac {\log {\left (\sqrt {4 y{\left (x \right )} + 1} + 1 \right )}}{2} = C_{1}, \ - \frac {\sqrt {4 y{\left (x \right )} + 1}}{2} + \frac {\log {\left (x \right )}}{2} - \frac {\log {\left (\sqrt {4 y{\left (x \right )} + 1} - 1 \right )}}{2} = C_{1}\right ] \]