Internal problem ID [3608]
Internal file name [OUTPUT/3101_Sunday_June_05_2022_08_52_09_AM_11956308/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 12
Problem number: 352.
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.
\[ \boxed {y^{\prime } x^{3}-\cos \left (y\right ) \left (\cos \left (y\right )-2 \sin \left (y\right ) x^{2}\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x^{3}-\cos \left (y\right ) \left (\cos \left (y\right )-2 \sin \left (y\right ) x^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\cos \left (y\right ) \left (\cos \left (y\right )-2 \sin \left (y\right ) x^{2}\right )}{x^{3}} \end {array} \]
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, 1/x^2*(1+cos(2*y))]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 15
dsolve(x^3*diff(y(x),x) = cos(y(x))*(cos(y(x))-2*x^2*sin(y(x))),y(x), singsol=all)
\[ y \left (x \right ) = \arctan \left (\frac {\ln \left (x \right )-c_{1}}{x^{2}}\right ) \]
✓ Solution by Mathematica
Time used: 5.952 (sec). Leaf size: 55
DSolve[x^3 y'[x]==Cos[y[x]](Cos[y[x]]-2 x^2 Sin[y[x]]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arctan \left (\frac {\log (x)+4 c_1}{x^2}\right ) \\ y(x)\to -\frac {1}{2} \pi \sqrt {\frac {1}{x^4}} x^2 \\ y(x)\to \frac {1}{2} \pi \sqrt {\frac {1}{x^4}} x^2 \\ \end{align*}