66.1.5 problem Problem 5

Internal problem ID [13781]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 5
Date solved : Monday, March 31, 2025 at 08:11:37 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 49

ode:=y(x)-x*diff(y(x),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}}}{c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.514 (sec). Leaf size: 68

ode=y[x]-x*D[y[x],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+\sqrt {\frac {1}{x^2}} x \sqrt {1+c_1 x^2}}{x} \\ 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: 0.710 (sec). Leaf size: 32

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)
 

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