77.1.23 problem 39 (page 41)

Internal problem ID [17913]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 39 (page 41)
Date solved : Tuesday, January 28, 2025 at 11:12:18 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: 0.048 (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: 0.107 (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*}