62.15.2 problem Ex 2

Internal problem ID [12810]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 26. Equations solvable for \(x\). Page 55
Problem number : Ex 2
Date solved : Wednesday, March 05, 2025 at 08:33:06 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.142 (sec). Leaf size: 51

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

\begin{align*} y &= -\frac {x}{a} \\ y &= \frac {x}{a} \\ y &= 0 \\ y &= {\mathrm e}^{\operatorname {RootOf}\left ({\mathrm e}^{2 \textit {\_Z}} \sinh \left (-\textit {\_Z} +c_{1} -\ln \left (x \right )\right )^{2} a^{2}+1\right )} x \\ \end{align*}

✓ Mathematica. Time used: 2.746 (sec). Leaf size: 89

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

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

✗ Sympy

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

Timed Out