Internal problem ID [11190]
Internal file name [OUTPUT/10176_Saturday_December_03_2022_08_03_25_AM_40769649/index.tex]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter 2, differential equations of the first order and the first degree. Article 19.
Summary. Page 29
Problem number: Ex 22.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right )-\left (x^{2}+y^{2}+x \right ) \left (y^{\prime } x -y\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (x^{2}+y^{2}\right ) \left (x +y y^{\prime }\right )-\left (x^{2}+y^{2}+x \right ) \left (y^{\prime } x -y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {\left (x^{2}+y^{2}\right ) x +\left (x^{2}+y^{2}+x \right ) y}{\left (x^{2}+y^{2}\right ) y-\left (x^{2}+y^{2}+x \right ) x} \end {array} \]
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 = 2`[0, (x^4+2*y^2*x^2+y^4+1/2*x^3-1/2*y*x^2+1/2*y^2*x-1/2*y^3)/(x^3-x^2*y+x*y^2-y^
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 45
dsolve((x^2+y(x)^2)*(x+y(x)*diff(y(x),x))=(x^2+y(x)^2+x)*(x*diff(y(x),x)-y(x)),y(x), singsol=all)
\[ y \left (x \right ) = -\cot \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +2 \ln \left (2 \csc \left (\textit {\_Z} \right )^{2} x^{2}+\cot \left (\textit {\_Z} \right ) x +x \right )-\ln \left (\csc \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \]
✓ Solution by Mathematica
Time used: 0.548 (sec). Leaf size: 53
DSolve[(x^2+y[x]^2)*(x+y[x]*y'[x])==(x^2+y[x]^2+x)*(x*y'[x]-y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \arctan \left (\frac {x}{y(x)}\right )-\frac {1}{4} \log \left (x^2+y(x)^2\right )+\frac {1}{2} \log \left (2 x^2+2 y(x)^2-y(x)+x\right )=c_1,y(x)\right ] \]