73.7.6 problem 6

Internal problem ID [15093]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 6
Date solved : Monday, March 31, 2025 at 01:23:28 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

ode:=x*y(x)*diff(y(x),x)-y(x)^2 = (x^4+x^2*y(x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {x^{2}+y^{2}}{\sqrt {x^{2} \left (x^{2}+y^{2}\right )}}+\ln \left (x \right )-c_1 = 0 \]

✓ Mathematica. Time used: 0.292 (sec). Leaf size: 54

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

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

✓ Sympy. Time used: 3.651 (sec). Leaf size: 49

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

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