76.23.7 problem Ex. 8

Internal problem ID [20148]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IV. Singular solutions. problems on chapter IV. page 49
Problem number : Ex. 8
Date solved : Thursday, October 02, 2025 at 05:33:16 PM
CAS classification : [_quadrature]

\begin{align*} 4 x \left (x -1\right ) \left (x -2\right ) {y^{\prime }}^{2}-\left (3 x^{2}-6 x +2\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 61

ode:=4*x*(x-1)*(x-2)*diff(y(x),x)^2-(3*x^2-6*x+2)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\int \frac {3 x^{2}-6 x +2}{\sqrt {x \left (x -1\right ) \left (x -2\right )}}d x}{2}+c_1 \\ y &= \frac {\int \frac {3 x^{2}-6 x +2}{\sqrt {x \left (x -1\right ) \left (x -2\right )}}d x}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 53

ode=4*x*(x-1)*(x-2)*D[y[x],x]^2-(3*x^2-6*x+2)^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1-i \sqrt {-x \left (x^2-3 x+2\right )}\\ y(x)&\to i \sqrt {-x \left (x^2-3 x+2\right )}+c_1 \end{align*}

✓ Sympy. Time used: 4.089 (sec). Leaf size: 129

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*(x - 2)*(x - 1)*Derivative(y(x), x)**2 - (3*x**2 - 6*x + 2)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - \int \sqrt {\frac {1}{x \left (x^{2} - 3 x + 2\right )}}\, dx + 3 \int x \sqrt {\frac {1}{x \left (x^{2} - 3 x + 2\right )}}\, dx - \frac {3 \int x^{2} \sqrt {\frac {1}{x \left (x^{2} - 3 x + 2\right )}}\, dx}{2}, \ y{\left (x \right )} = C_{1} + \int \sqrt {\frac {1}{x \left (x^{2} - 3 x + 2\right )}}\, dx - 3 \int x \sqrt {\frac {1}{x \left (x^{2} - 3 x + 2\right )}}\, dx + \frac {3 \int x^{2} \sqrt {\frac {1}{x \left (x^{2} - 3 x + 2\right )}}\, dx}{2}\right ] \]