29.8.14 problem 219

Internal problem ID [4819]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 8
Problem number : 219
Date solved : Tuesday, January 28, 2025 at 02:39:54 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Solution by Maple

Time used: 0.137 (sec). Leaf size: 33

dsolve(x*diff(y(x),x)+n*y(x) = f(x)*g(x^n*y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.508 (sec). Leaf size: 41

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

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