54.1.371 problem 380

Internal problem ID [11685]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 380
Date solved : Tuesday, September 30, 2025 at 10:07:42 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 642

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

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

✓ Mathematica. Time used: 60.059 (sec). Leaf size: 931

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

\begin{align*} y(x)&\to \frac {1}{4} \left (-x^2+\frac {x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{72} \left (-18 x^2-\frac {9 i \left (\sqrt {3}-i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+9 i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{72} \left (-18 x^2+\frac {9 i \left (\sqrt {3}+i\right ) x \left (x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}-9 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6+20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (-x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{4} \left (-x^2+\frac {x \left (x^3-8 e^{3 c_1}\right )}{\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{72} \left (-18 x^2+\frac {9 \left (1+i \sqrt {3}\right ) x \left (-x^3+8 e^{3 c_1}\right )}{\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}+9 i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right )\\ y(x)&\to \frac {1}{72} \left (-18 x^2+\frac {9 i \left (\sqrt {3}+i\right ) x \left (x^3-8 e^{3 c_1}\right )}{\sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}}-9 \left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6-20 e^{3 c_1} x^3+8 \sqrt {e^{3 c_1} \left (x^3+e^{3 c_1}\right ){}^3}+8 e^{6 c_1}}\right ) \end{align*}

✗ Sympy

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

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