77.23.2 problem 2

Internal problem ID [20568]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 2
Date solved : Thursday, October 02, 2025 at 06:09:01 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 33

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

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

✗ Mathematica

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

{}

✓ Sympy. Time used: 1.218 (sec). Leaf size: 82

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

\[ \left [ \log {\left (x \right )} = C_{1} + \begin {cases} - \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}, \ \log {\left (x \right )} = C_{1} + \begin {cases} \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}\right ] \]