83.46.5 problem Ex 5 page 70

Internal problem ID [19499]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter V. Singular solutions
Problem number : Ex 5 page 70
Date solved : Monday, March 31, 2025 at 07:26:13 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 85

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

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

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 55

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

\begin{align*} y(x)\to c_1-i \sqrt {x (a-x) (x-b)} \\ y(x)\to i \sqrt {x (a-x) (x-b)}+c_1 \\ \end{align*}

✓ Sympy. Time used: 1.403 (sec). Leaf size: 207

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

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