Internal problem ID [8548]
Internal file name [OUTPUT/7481_Sunday_June_05_2022_10_57_39_PM_93606449/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 212.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }+f \left (x^{2}+y^{2}\right ) g \left (x \right )=-x} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }+f \left (x^{2}+y^{2}\right ) g \left (x \right )=-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {f \left (x^{2}+y^{2}\right ) g \left (x \right )+x}{y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE 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/y*f(x^2+y^2)]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 30
dsolve(y(x)*diff(y(x),x)+f(x^2+y(x)^2)*g(x)+x=0,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}}{f \left (\textit {\_a}^{2}+x^{2}\right )}d \textit {\_a} +\int g \left (x \right )d x -c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.267 (sec). Leaf size: 95
DSolve[y[x]*y'[x]+f[x^2+y[x]^2]*g[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{f\left (x^2+K[2]^2\right )}-\int _1^x-\frac {2 K[1] K[2] f'\left (K[1]^2+K[2]^2\right )}{f\left (K[1]^2+K[2]^2\right )^2}dK[1]\right )dK[2]+\int _1^x\left (g(K[1])+\frac {K[1]}{f\left (K[1]^2+y(x)^2\right )}\right )dK[1]=c_1,y(x)\right ] \]