83.43.15 problem Ex 16 page 18

Internal problem ID [19452]
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 II. Equations of first order and first degree
Problem number : Ex 16 page 18
Date solved : Monday, March 31, 2025 at 07:15:14 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 111

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

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

✓ Mathematica. Time used: 0.521 (sec). Leaf size: 109

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

\begin{align*} y(x)\to -x+\sqrt [3]{3 a^2 x+x^3+3 c_1} \\ y(x)\to -x+\frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{3 a^2 x+x^3+3 c_1} \\ y(x)\to -x-\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [3]{3 a^2 x+x^3+3 c_1} \\ \end{align*}

✓ Sympy. Time used: 2.744 (sec). Leaf size: 78

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

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