83.23.12 problem 12

Internal problem ID [19222]
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 : 12
Date solved : Monday, March 31, 2025 at 06:57:45 PM
CAS classification : [_quadrature]

\begin{align*} x {y^{\prime }}^{2}-\left (x -a \right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 43

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

\begin{align*} y(x)\to -\frac {2}{3} \sqrt {x} (x-3 a)+c_1 \\ y(x)\to \frac {2}{3} \sqrt {x} (x-3 a)+c_1 \\ \end{align*}

✓ Sympy. Time used: 0.517 (sec). Leaf size: 41

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

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