75.3.23 problem 23

Internal problem ID [19927]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 23
Date solved : Thursday, October 02, 2025 at 05:02:22 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} 5 x y y^{\prime }-x^{2}-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=5*x*y(x)*diff(y(x),x)-x^2-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {x^{{2}/{5}} \left (4 c_1 +x^{{8}/{5}}\right )}}{2} \\ y &= \frac {\sqrt {x^{{2}/{5}} \left (4 c_1 +x^{{8}/{5}}\right )}}{2} \\ \end{align*}

✓ Mathematica. Time used: 3.406 (sec). Leaf size: 50

ode=5*x*y[x]*D[y[x],x]-(x^2+y[x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} \sqrt {x^2+4 c_1 x^{2/5}}\\ y(x)&\to \sqrt {\frac {x^2}{4}+c_1 x^{2/5}} \end{align*}

✓ Sympy. Time used: 0.282 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + 5*x*y(x)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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