60.1.127 problem 128

Internal problem ID [10141]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 128
Date solved : Tuesday, January 28, 2025 at 04:26:00 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} x y^{\prime }+a y-f \left (x \right ) g \left (x^{a} y\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.128 (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)
 

\[ y = \operatorname {RootOf}\left (-\int x^{a -1} fd x +\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x^{-a} \]

✓ Solution by Mathematica

Time used: 0.552 (sec). Leaf size: 41

DSolve[x*D[y[x],x] + a*y[x] - f[x]*g[x^a*y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{x^a y(x)}\frac {1}{g(K[1])}dK[1]=\int _1^xf(K[2]) K[2]^{a-1}dK[2]+c_1,y(x)\right ] \]