81.3.26 problem 26

Internal problem ID [18565]
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 : 26
Date solved : Thursday, March 13, 2025 at 12:18:57 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} 5 x y y^{\prime }-4 x^{2}-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 31

ode:=5*x*y(x)*diff(y(x),x)-4*x^2-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= \sqrt {\left (x^{{8}/{5}}+c_{1} \right ) x^{{2}/{5}}} \\ y \left (x \right ) &= -\sqrt {\left (x^{{8}/{5}}+c_{1} \right ) x^{{2}/{5}}} \\ \end{align*}

✓ Mathematica. Time used: 3.454 (sec). Leaf size: 43

ode=(5*x*y[x])*D[y[x],x]-(4*x^2+y[x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.393 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2 + 5*x*y(x)*Derivative(y(x), x) - y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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