Internal problem ID [5261]
Internal file name [OUTPUT/4752_Friday_February_02_2024_05_11_34_AM_63088358/index.tex]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 4. Equations of first order and first degree (Variable separable). Supplemetary
problems. Page 22
Problem number: 52.
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)]`]]
Unable to solve or complete the solution.
\[ \boxed {-2 \sin \left (y\right )+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime }=-x -3} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -2 \sin \left (y\right )+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime }=-x -3 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x +2 \sin \left (y\right )-3}{\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right )} \end {array} \]
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`[0, (-3+8*sin(y)-4*x)/(-2*x+4*sin(y)+3)/cos(y)]
✓ Solution by Maple
Time used: 0.203 (sec). Leaf size: 22
dsolve((x-2*sin(y(x))+3)+(2*x-4*sin(y(x))-3)*cos(y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \arcsin \left (\frac {9 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-\frac {1}{3}-\frac {8 x}{9}}}{9}\right )}{8}+\frac {3}{8}+\frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 60.95 (sec). Leaf size: 73
DSolve[(x-2*Sin[y[x]]+3)+(2*x-4*Sin[y[x]]-3)*Cos[y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arcsin \left (\frac {1}{8} \left (9 W\left (-\frac {1}{9} e^{-\frac {2}{9} (4 x+3-8 c_1)}\right )+4 x+3\right )\right ) \\ y(x)\to \arcsin \left (\frac {1}{8} \left (9 W\left (-\frac {1}{9} e^{-\frac {2}{9} (4 x+3-8 c_1)}\right )+4 x+3\right )\right ) \\ \end{align*}