Internal problem ID [2046]
Internal file name [OUTPUT/2046_Sunday_February_25_2024_06_47_09_AM_19711343/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 12, page 46
Problem number: 14.
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(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+y \ln \left (y\right ) \tan \left (x \right )-2 y=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y \ln \left (y\right ) \tan \left (x \right )-2 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y \ln \left (y\right ) \tan \left (x \right )+2 y \end {array} \]
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 trying an equivalence to an Abel ODE 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)] <- symmetry pattern of the form [0, F(x)*G(y)] successful`
✓ Solution by Maple
Time used: 1.25 (sec). Leaf size: 25
dsolve(diff(y(x),x)+y(x)*ln(y(x))*tan(x)=2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (-\frac {\cos \left (x \right )}{\sin \left (x \right )-1}\right )^{2 \cos \left (x \right )} {\mathrm e}^{\cos \left (x \right ) c_{1}} \]
✓ Solution by Mathematica
Time used: 2.002 (sec). Leaf size: 17
DSolve[y'[x]+y[x]*Log[y[x]]*Tan[x]==2*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{2 \cos (x) \left (\coth ^{-1}(\sin (x))+c_1\right )} \]