29.28.6 problem 804

Internal problem ID [5387]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 28
Problem number : 804
Date solved : Sunday, March 30, 2025 at 08:04:13 AM
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: 0.420 (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: 4.727 (sec). Leaf size: 78

ode=(D[y[x],x])^2+a*x^3*D[y[x],x]-2*a*x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to 8 e^{4 c_1}-2 \sqrt {a} e^{2 c_1} x^2 \\ y(x)\to \frac {1}{8} e^{2 c_1} \left (2 \sqrt {a} x^2+e^{2 c_1}\right ) \\ y(x)\to 0 \\ y(x)\to -\frac {a x^4}{8} \\ \end{align*}

✓ Sympy. Time used: 2.308 (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)
 

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