62.12.14 problem Ex 15

Internal problem ID [12781]
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 : Wednesday, March 05, 2025 at 08:30:26 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.273 (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: 0.232 (sec). Leaf size: 124

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]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {2 y(x)^2}{5 \left (K[1] y(x)^2-1\right )}+\frac {4}{5 K[1]}\right )dK[1]+\int _1^{y(x)}\left (\frac {4 x K[2]}{5 \left (x K[2]^2-1\right )}-\int _1^x\left (\frac {4 K[2]}{5 \left (K[1] K[2]^2-1\right )}-\frac {4 K[1] K[2]^3}{5 \left (K[1] K[2]^2-1\right )^2}\right )dK[1]-\frac {2}{5 K[2]}\right )dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 11.038 (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)
 

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