Internal problem ID [12466]
Internal file name [OUTPUT/11119_Monday_October_16_2023_09_49_42_PM_22236623/index.tex]
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.
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.
\[ \boxed {y^{\prime } \left (x^{2} y^{3}+x y\right )=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \left (x^{2} y^{3}+x y\right )=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{x^{2} y^{3}+x y} \end {array} \]
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(diff(y(x),x)*(x^2*y(x)^3+x*y(x))=1,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: 0.18 (sec). Leaf size: 76
DSolve[y'[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*}