54.2.178 problem 755

Internal problem ID [12052]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 755
Date solved : Wednesday, October 01, 2025 at 12:07:51 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} y^{\prime }&=\frac {y^{{3}/{2}}}{y^{{3}/{2}}+x^{2}-2 x y+y^{2}} \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 71

ode:=diff(y(x),x) = y(x)^(3/2)/(y(x)^(3/2)+x^2-2*x*y(x)+y(x)^2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {4 \sqrt {y}\, x^{2}-y^{{7}/{2}} c_1 +\left (2 c_1 x +4\right ) y^{{5}/{2}}+4 y^{2}+\left (-c_1 \,x^{2}-8 x +1\right ) y^{{3}/{2}}-4 x y}{\left (x -y\right )^{2} y^{{3}/{2}}} = 0 \]

✓ Mathematica. Time used: 60.183 (sec). Leaf size: 2213

ode=D[y[x],x] == y[x]^(3/2)/(x^2 - 2*x*y[x] + y[x]^(3/2) + y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - y(x)**(3/2)/(x**2 - 2*x*y(x) + y(x)**(3/2) + y(x)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)**(3/2)/(x**2 - 2*x*y(x) + y(x)**(3/2) + y(x)**2) cannot be solved by the lie group method