Internal problem ID [7707]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 127.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {x y^{\prime }-y f \left (x^{a} y^{b}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.218 (sec). Leaf size: 39
dsolve(x*diff(y(x),x) - y(x)*f(x^a*y(x)^b)=0,y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \relax (x )}\frac {1}{\left (f \left (x^{a} \textit {\_a}^{b}\right ) b +a \right ) \textit {\_a}}d \textit {\_a} -\frac {\ln \relax (x )}{b}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.344 (sec). Leaf size: 186
DSolve[x*y'[x] - y[x]*f[x^a*y[x]^b]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {b}{\left (a+b f\left (x^a K[2]^b\right )\right ) K[2]}-\int _1^x\left (\frac {b^2 K[1]^{a-1} K[2]^{b-1} f'\left (K[1]^a K[2]^b\right )}{a+b f\left (K[1]^a K[2]^b\right )}-\frac {b^3 f\left (K[1]^a K[2]^b\right ) K[1]^{a-1} K[2]^{b-1} f'\left (K[1]^a K[2]^b\right )}{\left (a+b f\left (K[1]^a K[2]^b\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b f\left (K[1]^a y(x)^b\right )}{\left (a+b f\left (K[1]^a y(x)^b\right )\right ) K[1]}dK[1]=c_1,y(x)\right ] \]