Internal problem ID [8464]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 128.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right )=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 33
dsolve(x*diff(y(x),x) + a*y(x) - f(x)*g(x^a*y(x))=0,y(x), singsol=all)
\[ y \left (x \right ) = \operatorname {RootOf}\left (-\left (\int f \left (x \right ) x^{a -1}d x \right )+\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x^{-a} \]
✓ Solution by Mathematica
Time used: 1.892 (sec). Leaf size: 41
DSolve[x*y'[x] + a*y[x] - f[x]*g[x^a*y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{x^a y(x)}\frac {1}{g(K[1])}dK[1]=\int _1^xf(K[2]) K[2]^{a-1}dK[2]+c_1,y(x)\right ] \]