22.16 problem 624

Internal problem ID [3872]
Internal file name [OUTPUT/3365_Sunday_June_05_2022_09_12_28_AM_58937740/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 22
Problem number: 624.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[`y=_G(x,y')`]

Unable to solve or complete the solution.

\[ \boxed {\left (\cot \left (x \right )-2 y^{2}\right ) y^{\prime }-y^{3} \csc \left (x \right ) \sec \left (x \right )=0} \] Unable to determine ODE type.

22.16.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\cot \left (x \right )-2 y^{2}\right ) y^{\prime }-y^{3} \csc \left (x \right ) \sec \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{3} \csc \left (x \right ) \sec \left (x \right )}{\cot \left (x \right )-2 y^{2}} \end {array} \]

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 
trying an equivalence to an Abel ODE 
equivalence obtained to this Abel ODE: diff(y(x),x) = 2*(cot(x)*csc(x)*sec(x)+cot(x)^2+1)/cot(x)^2*y(x)^2+4/cot(x)^2*csc(x)*sec(x)*y 
trying to solve the Abel ODE ... 
Looking for potential symmetries 
Looking for potential symmetries 
differential order: 1; trying a linearization to 2nd order 
--- trying a change of variables {x -> y(x), y(x) -> x} 
differential order: 1; trying a linearization to 2nd order 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
`, `-> Computing symmetries using: way = 3 
`, `-> Computing symmetries using: way = 4 
`, `-> Computing symmetries using: way = 5 
trying symmetry patterns for 1st order ODEs 
-> trying a symmetry pattern of the form [F(x)*G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)*G(y)] 
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
`, `-> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE`, diff(y(x), x)+y(x)*(cot(x)*tan(x)+1)/cot(x), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE`, diff(y(x), x)-y(x)*(cot(x)-tan(x)), y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
`, `-> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE`, diff(y(x), x)-3*y(x)/x, y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE`, diff(y(x), x)-y(x)/x, y(x)`      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> trying a symmetry pattern of the form [F(x)+G(y), 0] 
-> trying a symmetry pattern of the form [0, F(x)+G(y)] 
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
-> trying a symmetry pattern of conformal type`
 

✗ Solution by Maple

dsolve((cot(x)-2*y(x)^2)*diff(y(x),x) = y(x)^3*csc(x)*sec(x),y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 7.946 (sec). Leaf size: 74

DSolve[(Cot[x]-2 y[x]^2)y'[x]==y[x]^3 Csc[x] Sec[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {i \sqrt {\cot (x)} \sqrt {W\left (-2 e^{-8 c_1} \tan (x)\right )}}{\sqrt {2}} \\ y(x)\to \frac {i \sqrt {\cot (x)} \sqrt {W\left (-2 e^{-8 c_1} \tan (x)\right )}}{\sqrt {2}} \\ y(x)\to 0 \\ \end{align*}