47.1.17 problem 17

Internal problem ID [9737]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 06:32:33 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} x y \left (x^{2}+y^{2}\right ) \left ({y^{\prime }}^{2}-1\right )&=y^{\prime } \left (x^{4}+x^{2} y^{2}+y^{4}\right ) \end{align*}

✓ Maple. Time used: 0.275 (sec). Leaf size: 142

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

\begin{align*} y &= \frac {\sqrt {-c_1 \,x^{2}-\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= \frac {\sqrt {-c_1 \,x^{2}+\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {-c_1 \,x^{2}-\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= -\frac {\sqrt {-c_1 \,x^{2}+\sqrt {c_1^{2} x^{4}+1}}}{\sqrt {c_1}} \\ y &= \sqrt {2 \ln \left (x \right )+c_1}\, x \\ y &= -\sqrt {2 \ln \left (x \right )+c_1}\, x \\ \end{align*}

✓ Mathematica. Time used: 0.325 (sec). Leaf size: 78

ode=x*y[x]*(x^2+y[x]^2)*((D[y[x],x])^2-1)==D[y[x],x]*(x^4+x^2*y[x]^2+y[x]^4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \sqrt {2 \log (x)+c_1}\\ y(x)&\to x \sqrt {2 \log (x)+c_1}\\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2+1}{K[1] \left (K[1]^2+2\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \end{align*}

✓ Sympy. Time used: 5.049 (sec). Leaf size: 104

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + y(x)**2)*(Derivative(y(x), x)**2 - 1)*y(x) - (x**4 + x**2*y(x)**2 + y(x)**4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - x \sqrt {C_{1} + 2 \log {\left (x \right )}}, \ y{\left (x \right )} = x \sqrt {C_{1} + 2 \log {\left (x \right )}}, \ y{\left (x \right )} = - \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} - \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + \sqrt {C_{1} + x^{4}}}\right ] \]