69.1.50 problem 69

Internal problem ID [14124]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 69
Date solved : Wednesday, March 05, 2025 at 10:35:49 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} y^{\prime } \left (x^{2} y^{3}+x y\right )&=1 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 78

ode:=diff(y(x),x)*(x^2*y(x)^3+x*y(x)) = 1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.113 (sec). Leaf size: 76

ode=D[y[x],x]*(x^2*y[x]^3+x*y[x])==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\sqrt {2 x W\left (c_1 e^{\frac {1}{2 x}-1}\right )+2 x-1}}{\sqrt {x}} \\ y(x)\to \frac {\sqrt {2 x W\left (c_1 e^{\frac {1}{2 x}-1}\right )+2 x-1}}{\sqrt {x}} \\ \end{align*}

✓ Sympy. Time used: 1.264 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2*y(x)**3 + x*y(x))*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} - \left (y^{2}{\left (x \right )} - 2\right ) \sqrt {e^{y^{2}{\left (x \right )}}} - \frac {\sqrt {e^{y^{2}{\left (x \right )}}}}{x} = 0 \]