Internal problem ID [3558]
Internal file name [OUTPUT/3051_Sunday_June_05_2022_08_50_41_AM_90702269/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 11
Problem number: 302.
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.
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}+1\right ) y^{\prime }+x \sin \left (y\right ) \cos \left (y\right )-x \left (x^{2}+1\right ) \cos \left (y\right )^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-x \sin \left (y\right ) \cos \left (y\right )+x \left (x^{2}+1\right ) \cos \left (y\right )^{2}}{x^{2}+1} \end {array} \]
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: 0.047 (sec). Leaf size: 142
dsolve((x^2+1)*diff(y(x),x)+x*sin(y(x))*cos(y(x)) = x*(x^2+1)*cos(y(x))^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\arctan \left (\frac {6 \sqrt {x^{2}+1}\, \left (x^{2} \sqrt {x^{2}+1}+\sqrt {x^{2}+1}+3 c_{1} \right )}{10+6 c_{1} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}, \frac {8+6 \left (-x^{2}-1\right ) c_{1} \sqrt {x^{2}+1}-x^{6}-3 x^{4}+6 x^{2}-9 c_{1}^{2}}{10+6 c_{1} \left (x^{2}+1\right )^{\frac {3}{2}}+x^{6}+3 x^{4}+12 x^{2}+9 c_{1}^{2}}\right )}{2} \]
✓ Solution by Mathematica
Time used: 8.716 (sec). Leaf size: 97
DSolve[(1+x^2)y'[x]+x Sin[y[x]] Cos[y[x]]==x(1+x^2) (Cos[y[x]])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arctan \left (\frac {x^4+2 x^2-6 c_1 \sqrt {x^2+1}+1}{3 x^2+3}\right ) \\ y(x)\to -\frac {1}{2} \pi \sqrt {\frac {1}{x^2+1}} \sqrt {x^2+1} \\ y(x)\to \frac {1}{2} \pi \sqrt {\frac {1}{x^2+1}} \sqrt {x^2+1} \\ \end{align*}