75.5.19 problem 118

Internal problem ID [16684]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 5. Homogeneous equations. Exercises page 44
Problem number : 118
Date solved : Monday, March 31, 2025 at 03:05:26 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} y \left (1+\sqrt {x^{2} y^{4}+1}\right )+2 x y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=y(x)*(1+(x^2*y(x)^4+1)^(1/2))+2*x*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_1 -2 \int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {\textit {\_a}^{4}+1}}d \textit {\_a} \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.938 (sec). Leaf size: 80

ode=y[x]*(1+Sqrt[x^2*y[x]^4+1])+2*x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {i \sqrt {2} e^{\frac {c_1}{2}}}{\sqrt {-x^2+e^{2 c_1}}} \\ y(x)\to \frac {i \sqrt {2} e^{\frac {c_1}{2}}}{\sqrt {-x^2+e^{2 c_1}}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 35.529 (sec). Leaf size: 32

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

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