Internal problem ID [15062]
Internal file name [OUTPUT/15063_Sunday_April_21_2024_01_22_02_PM_29534322/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page
54
Problem number: 170.
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.
\[ \boxed {y^{\prime }+x \sin \left (2 y\right )-2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+x \sin \left (2 y\right )-2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-x \sin \left (2 y\right )+2 x \,{\mathrm e}^{-x^{2}} \cos \left (y\right )^{2} \end {array} \]
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 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, exp(-x^2)*(1+cos(2*y))], [0, exp(-x^2)*cos(2*y)*x^2+exp(-x^2)*x^2-sin(2*y)]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 19
dsolve(diff(y(x),x)+x*sin(2*y(x))=2*x*exp(-x^2)*cos(y(x))^2,y(x), singsol=all)
\[ y \left (x \right ) = \arctan \left (\left (x^{2}+2 c_{1} \right ) {\mathrm e}^{-x^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 10.038 (sec). Leaf size: 70
DSolve[y'[x]+x*Sin[2*y[x]]==2*x*Exp[-x^2]*Cos[y[x]]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arctan \left (e^{-x^2} \left (x^2+c_1\right )\right ) \\ y(x)\to -\frac {1}{2} \pi e^{x^2} \sqrt {e^{-2 x^2}} \\ y(x)\to \frac {1}{2} \pi e^{x^2} \sqrt {e^{-2 x^2}} \\ \end{align*}