Internal problem ID [8656]
Internal file name [OUTPUT/7589_Sunday_June_05_2022_11_08_42_PM_96580843/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 320.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
Unable to solve or complete the solution.
\[ \boxed {\left (y^{3} x^{2}+y x \right ) y^{\prime }=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y^{3} x^{2}+y x \right ) y^{\prime }=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{y^{3} x^{2}+y x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati <- Bernoulli successful <- inverse_Riccati successful`
✓ Solution by Maple
Time used: 0.031 (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 \left (x \right ) &= \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 \left (x \right ) &= -\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[-1 + (x*y[x] + x^2*y[x]^3)*y'[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*}