Internal problem ID [4995]
Internal file name [OUTPUT/4488_Sunday_June_05_2022_02_57_44_PM_99596253/index.tex]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston.
Pearson 2018.
Section: Chapter 2, First order differential equations. Section 2.4, Exact equations. Exercises.
page 64
Problem number: 18.
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.
\[ \boxed {y^{2}-\cos \left (x +y\right )-\left (2 y x -\cos \left (x +y\right )-{\mathrm e}^{y}\right ) y^{\prime }=-2 x} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{2}-\cos \left (x +y\right )-\left (2 y x -\cos \left (x +y\right )-{\mathrm e}^{y}\right ) y^{\prime }=-2 x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-2 x -y^{2}+\cos \left (x +y\right )}{2 y x -\cos \left (x +y\right )-{\mathrm e}^{y}} \end {array} \]
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
dsolve((2*x+y(x)^2-cos(x+y(x)))-(2*x*y(x)-cos(x+y(x))-exp(y(x)))*diff(y(x),x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[(2*x+y[x]^2-Cos[x+y[x]])-(2*x*y[x]-Cos[x+y[x]]-Exp[y[x]])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved