54.2.95 problem 671

Internal problem ID [11969]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 671
Date solved : Sunday, October 12, 2025 at 02:07:29 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 207

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

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

✓ Mathematica. Time used: 0.15 (sec). Leaf size: 144

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 K[2] x^2}{2 x^2 K[2]^4+4 x K[2]^2-x^2+2}-\int _1^x-\frac {8 K[1]^2 K[2]^3+8 K[1] K[2]}{\left (2 K[1]^2 K[2]^4+4 K[1] K[2]^2-K[1]^2+2\right )^2}dK[1]\right )dK[2]+\int _1^x\left (\frac {1}{K[1]^2}+\frac {1}{2 K[1]^2 y(x)^4+4 K[1] y(x)^2-K[1]^2+2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*y(x)**4 + 2*x*y(x)**2 + 1)/(x**4*y(x)) cannot be solved by the factorable group method