Internal problem ID [2968]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 8
Problem number: 220.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {y^{\prime } x -y f \left (x^{m} y^{n}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 39
dsolve(x*diff(y(x),x) = y(x)*f(x^m*y(x)^n),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \relax (x )}\frac {1}{\left (f \left (x^{m} \textit {\_a}^{n}\right ) n +m \right ) \textit {\_a}}d \textit {\_a} -\frac {\ln \relax (x )}{n}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.215 (sec). Leaf size: 186
DSolve[x y'[x]==y[x] f[x^m y[x]^n] ,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {n}{\left (m+n f\left (x^m K[2]^n\right )\right ) K[2]}-\int _1^x\left (\frac {n^2 K[1]^{m-1} K[2]^{n-1} f'\left (K[1]^m K[2]^n\right )}{m+n f\left (K[1]^m K[2]^n\right )}-\frac {n^3 f\left (K[1]^m K[2]^n\right ) K[1]^{m-1} K[2]^{n-1} f'\left (K[1]^m K[2]^n\right )}{\left (m+n f\left (K[1]^m K[2]^n\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {n f\left (K[1]^m y(x)^n\right )}{\left (m+n f\left (K[1]^m y(x)^n\right )\right ) K[1]}dK[1]=c_1,y(x)\right ] \]