60.1.319 problem 320

Internal problem ID [10333]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 320
Date solved : Monday, January 27, 2025 at 07:13:16 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (x^{2} y^{3}+y x \right ) y^{\prime }-1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 78

dsolve((x^2*y(x)^3+x*y(x))*diff(y(x),x)-1 = 0,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: 0.127 (sec). Leaf size: 76

DSolve[-1 + (x*y[x] + x^2*y[x]^3)*D[y[x],x]==0,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*}