56.12.14 problem Ex 15

Internal problem ID [14141]
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 19. Summary. Page 29
Problem number : Ex 15
Date solved : Thursday, October 02, 2025 at 09:16:56 AM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.239 (sec). Leaf size: 45

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

\begin{align*} y &= \frac {c_1 +\sqrt {4 x^{5}+c_1^{2}}}{2 x^{3}} \\ y &= \frac {c_1 -\sqrt {4 x^{5}+c_1^{2}}}{2 x^{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.64 (sec). Leaf size: 75

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

\begin{align*} y(x)&\to -\frac {\sqrt {4 x^5+e^{5 c_1}}+e^{\frac {5 c_1}{2}}}{2 x^3}\\ y(x)&\to \frac {\sqrt {4 x^5+e^{5 c_1}}-e^{\frac {5 c_1}{2}}}{2 x^3} \end{align*}

✓ Sympy. Time used: 7.788 (sec). Leaf size: 180

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

\[ \left [ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\frac {2 - \frac {\sqrt {4 x^{5} e^{C_{1}} + 1} e^{- C_{1}}}{x^{5}} + \frac {e^{- C_{1}}}{x^{5}}}{x}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\frac {2 - \frac {\sqrt {4 x^{5} e^{C_{1}} + 1} e^{- C_{1}}}{x^{5}} + \frac {e^{- C_{1}}}{x^{5}}}{x}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {\frac {2 + \frac {\sqrt {4 x^{5} e^{C_{1}} + 1} e^{- C_{1}}}{x^{5}} + \frac {e^{- C_{1}}}{x^{5}}}{x}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {\frac {2 + \frac {\sqrt {4 x^{5} e^{C_{1}} + 1} e^{- C_{1}}}{x^{5}} + \frac {e^{- C_{1}}}{x^{5}}}{x}}}{2}\right ] \]