54.7.156 problem 1773 (book 6.182)

Internal problem ID [13005]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1773 (book 6.182)
Date solved : Friday, October 03, 2025 at 03:58:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x^{2} \left (x -y\right ) y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 45

ode:=x^2*(x-y(x))*diff(diff(y(x),x),x)+a*(-y(x)+x*diff(y(x),x))^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x \\ y &= -\operatorname {RootOf}\left (c_1 a x \,\textit {\_Z}^{a}-c_1 x \,\textit {\_Z}^{a}-c_2 a \,\textit {\_Z}^{a}+c_2 \,\textit {\_Z}^{a}+\textit {\_Z} \,x^{a}\right )+x \\ \end{align*}

✓ Mathematica. Time used: 21.304 (sec). Leaf size: 285

ode=a*(-y[x] + x*D[y[x],x])^2 + x^2*(x - y[x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {a-1}{a x c_1 \left (\frac {K[3]}{x}-1\right )^a-x c_1 \left (\frac {K[3]}{x}-1\right )^a-x+K[3]}dK[3]-\int _1^x\left (\int _1^{y(x)}\frac {(a-1) \left (-\frac {a^2 c_1 K[3] \left (\frac {K[3]}{K[4]}-1\right )^{a-1}}{K[4]}+\frac {a c_1 K[3] \left (\frac {K[3]}{K[4]}-1\right )^{a-1}}{K[4]}+a c_1 \left (\frac {K[3]}{K[4]}-1\right )^a-c_1 \left (\frac {K[3]}{K[4]}-1\right )^a-1\right )}{\left (a c_1 K[4] \left (\frac {K[3]}{K[4]}-1\right )^a-c_1 K[4] \left (\frac {K[3]}{K[4]}-1\right )^a+K[3]-K[4]\right ){}^2}dK[3]-\frac {(a-1) \left (c_1 \left (\frac {y(x)}{K[4]}-1\right )^a-\frac {1-\frac {a y(x)}{K[4]}}{a-1}\right )}{a c_1 K[4] \left (\frac {y(x)}{K[4]}-1\right )^a-c_1 K[4] \left (\frac {y(x)}{K[4]}-1\right )^a-K[4]+y(x)}\right )dK[4]=c_2,y(x)\right ] \]

✗ Sympy

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

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