Internal problem ID [8417]
Internal file name [OUTPUT/7350_Sunday_June_05_2022_10_53_12_PM_68797936/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 80.
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.
\[ \boxed {y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )=f^{\prime }\left (x \right )+1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )=f^{\prime }\left (x \right )+1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-f \left (x \right ) \sin \left (y\right )-\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )+f^{\prime }\left (x \right )+1 \end {array} \]
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)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 41
dsolve(diff(y(x),x) + f(x)*sin(y(x)) + (1-diff(f(x),x))*cos(y(x)) - diff(f(x),x) - 1=0,y(x), singsol=all)
\[ y \left (x \right ) = -2 \arctan \left (\frac {{\mathrm e}^{\int f \left (x \right )d x}-\left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right ) f \left (x \right )-c_{1} f \left (x \right )}{c_{1} +\int {\mathrm e}^{\int f \left (x \right )d x}d x}\right ) \]
✓ Solution by Mathematica
Time used: 7.14 (sec). Leaf size: 68
DSolve[y'[x] + f[x]*Sin[y[x]] + (1-f'[x])*Cos[y[x]] - f'[x]- 1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 \arctan \left (f(x)+\frac {\exp \left (-\int _1^x-f(K[1])dK[1]\right )}{\int _1^x-\exp \left (-\int _1^{K[2]}-f(K[1])dK[1]\right )dK[2]+c_1}\right ) \\ y(x)\to 2 \arctan (f(x)) \\ \end{align*}