83.5.1 problem 1 (a)

Internal problem ID [21936]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. VI at page 47
Problem number : 1 (a)
Date solved : Thursday, October 02, 2025 at 08:16:38 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} x y^{\prime }-y&=x^{2} y y^{\prime } \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.318 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to \frac {1}{x}-\sqrt {\frac {1}{x^2}} \sqrt {1+c_1 x^2}\\ y(x)&\to \frac {1}{x}+\sqrt {\frac {1}{x^2}} \sqrt {1+c_1 x^2}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.480 (sec). Leaf size: 31

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

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