77.23.29 problem 29

Internal problem ID [20595]
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 : 29
Date solved : Thursday, October 02, 2025 at 06:14:59 PM
CAS classification : [_separable]

\begin{align*} 4 {y^{\prime }}^{2} x^{2} \left (x -1\right )-4 y^{\prime } x y \left (4 x -3\right )+\left (16 x -9\right ) y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 57

ode:=4*diff(y(x),x)^2*x^2*(x-1)-4*diff(y(x),x)*x*y(x)*(4*x-3)+(16*x-9)*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {c_1 \sqrt {1-x}\, \sqrt {x -1}\, x^{{3}/{2}}}{\sqrt {x}+1} \\ y &= \frac {c_1 \left (\sqrt {x}+1\right ) \sqrt {x -1}\, x^{{3}/{2}}}{\sqrt {1-x}} \\ \end{align*}

✓ Mathematica. Time used: 0.096 (sec). Leaf size: 44

ode=4*D[y[x],x]^2*x^2*(x-1)-4*D[y[x],x]*x*y[x]*(4*x-3)+(16*x-9)*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \left (\sqrt {x}+1\right ) x^{3/2}\\ y(x)&\to c_1 \left (\sqrt {x}-1\right ) x^{3/2}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 3.335 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x - 1)*Derivative(y(x), x)**2 - 4*x*(4*x - 3)*y(x)*Derivative(y(x), x) + (16*x - 9)*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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