Internal problem ID [3532]
Internal file name [OUTPUT/3025_Sunday_June_05_2022_08_49_59_AM_79187334/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 10
Problem number: 276.
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.
\[ \boxed {y^{\prime } x^{2}-\sec \left (y\right )-3 x \tan \left (y\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } x^{2}-\sec \left (y\right )-3 x \tan \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sec \left (y\right )+3 x \tan \left (y\right )}{x^{2}} \end {array} \]
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, 1/x*(4*tan(y)*x+sec(y))]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 17
dsolve(x^2*diff(y(x),x) = sec(y(x))+3*x*tan(y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \arcsin \left (\frac {c_{1} x^{4}-1}{4 x}\right ) \]
✓ Solution by Mathematica
Time used: 10.03 (sec). Leaf size: 23
DSolve[x^2 y'[x]==Sec[y[x]]+3 x Tan[y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\arcsin \left (\frac {1}{4 x}+3 c_1 x^3\right ) \]