56.15.2 problem Ex 2

Internal problem ID [14170]
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 : Thursday, October 02, 2025 at 09:18:02 AM
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: 0.065 (sec). Leaf size: 71

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 &= \operatorname {RootOf}\left (a^{2} x^{4} \textit {\_Z}^{4}-2 a^{2} x^{2} \textit {\_Z}^{2} {\mathrm e}^{2 c_1}+a^{2} {\mathrm e}^{4 c_1}+4 x^{2} {\mathrm e}^{2 c_1}\right ) x \\ \end{align*}

✓ Mathematica. Time used: 0.919 (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. Time used: 157.406 (sec). Leaf size: 704

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)
 

\[ \text {Solution too large to show} \]