85.33.47 problem 47

Internal problem ID [22670]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 47
Date solved : Thursday, October 02, 2025 at 09:03:31 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: 0.014 (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: 0.317 (sec). Leaf size: 63

ode=x*D[y[x],{x,1}]-y[x]==x^2*y[x]*D[y[x],{x,1}]; 
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: 0.471 (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)
 

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