54.1.377 problem 386

Internal problem ID [11691]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 386
Date solved : Tuesday, September 30, 2025 at 10:07:59 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}+a \,x^{3} y^{\prime }-2 a \,x^{2} y&=0 \end{align*}

✓ Maple. Time used: 0.360 (sec). Leaf size: 27

ode:=diff(y(x),x)^2+a*x^3*diff(y(x),x)-2*a*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {a \,x^{4}}{8} \\ y &= \frac {c_{1} \left (a \,x^{2}+2 c_{1} \right )}{a} \\ \end{align*}

✓ Mathematica. Time used: 0.494 (sec). Leaf size: 120

ode=-2*a*x^2*y[x] + a*x^3*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}{2} (\cosh (2 c_1)+\sinh (2 c_1)) \left (-i \sqrt {2} \sqrt {a} x^2+2 \cosh (2 c_1)+2 \sinh (2 c_1)\right )\\ y(x)&\to -\frac {1}{2} (\cosh (2 c_1)+\sinh (2 c_1)) \left (i \sqrt {2} \sqrt {a} x^2+2 \cosh (2 c_1)+2 \sinh (2 c_1)\right )\\ y(x)&\to 0\\ y(x)&\to -\frac {a x^4}{8} \end{align*}

✓ Sympy. Time used: 1.520 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \frac {C_{1} a \left (C_{1} + x^{2}\right )}{2} \]