62.10.5 problem Ex 6

Internal problem ID [12763]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 17. Other forms which Integrating factors can be found. Page 25
Problem number : Ex 6
Date solved : Wednesday, March 05, 2025 at 08:26:30 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.092 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {m -\sqrt {-4 c_{1}^{2} x^{2}-4 c_{1} x +m^{2}}}{2 c_{1}} \\ y &= \frac {m +\sqrt {-4 c_{1}^{2} x^{2}-4 c_{1} x +m^{2}}}{2 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 54

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {m K[1]^2-2 K[1]-m}{(m K[1]-1) \left (K[1]^2+1\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 3.062 (sec). Leaf size: 70

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

\[ \left [ y{\left (x \right )} = - \frac {m e^{C_{1}}}{2} - \frac {\sqrt {m^{2} e^{2 C_{1}} - 4 x^{2} + 4 x e^{C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {m e^{C_{1}}}{2} + \frac {\sqrt {m^{2} e^{2 C_{1}} - 4 x^{2} + 4 x e^{C_{1}}}}{2}\right ] \]