69.1.50 problem 69

Internal problem ID [14203]
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 : Tuesday, January 28, 2025 at 06:21:44 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Solution by Maple

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

dsolve(diff(y(x),x)*(x^2*y(x)^3+x*y(x))=1,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*}

✓ Solution by Mathematica

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

DSolve[D[y[x],x]*(x^2*y[x]^3+x*y[x])==1,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*}