Internal problem ID [2031]
Internal file name [OUTPUT/2031_Sunday_February_25_2024_06_45_37_AM_90454972/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 11, page 45
Problem number: 21.
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.
\[ \boxed {x y \left (1+x y^{2}\right ) y^{\prime }=-1} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x y \left (1+x y^{2}\right ) y^{\prime }=-1, y \left (1\right )=0\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {1}{x y \left (1+x y^{2}\right )} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (1\right )=0 \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \end {array} \]
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: 0.328 (sec). Leaf size: 66
dsolve([1+x*y(x)*(1+x*y(x)^2)*diff(y(x),x)=0,y(1) = 0],y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {-2 \left (\operatorname {LambertW}\left (-1, -\frac {3 \,{\mathrm e}^{-\frac {2 x +1}{2 x}}}{2}\right ) x +x +\frac {1}{2}\right ) x}}{x} \\ y \left (x \right ) &= -\frac {\sqrt {-2 \left (\operatorname {LambertW}\left (-1, -\frac {3 \,{\mathrm e}^{-\frac {2 x +1}{2 x}}}{2}\right ) x +x +\frac {1}{2}\right ) x}}{x} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[{1+x*y[x]*(1+x*y[x]^2)*y'[x]==0,{y[1]==0}},y[x],x,IncludeSingularSolutions -> True]
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