Internal problem ID [8149]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 571.
ODE order: 1.
ODE degree: 0.
CAS Maple gives this as type [`y=_G(x,y')`]
\[ \boxed {a \,x^{n} f \left (y^{\prime }\right )+y^{\prime } x -y=0} \]
✓ Solution by Maple
Time used: 0.609 (sec). Leaf size: 199
dsolve(a*x^n*f(diff(y(x),x))+x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ \left [y \left (\textit {\_T} \right ) = a {\left ({\left (-\frac {-c_{1} a n +\left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right ) n -\left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right )}{a f \left (\textit {\_T} \right ) n}\right )}^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}\right )}^{n} f \left (\textit {\_T} \right )+\textit {\_T} {\left (-\frac {-c_{1} a n +\left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right ) n -\left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right )}{a f \left (\textit {\_T} \right ) n}\right )}^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}, x \left (\textit {\_T} \right ) = {\left (-\frac {-c_{1} a n +\left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right ) n -\left (\int f \left (\textit {\_T} \right )^{-\frac {1}{n}}d \textit {\_T} \right )}{a f \left (\textit {\_T} \right ) n}\right )}^{\frac {1}{n -1}} f \left (\textit {\_T} \right )^{\frac {1}{n \left (n -1\right )}}\right ] \]
✓ Solution by Mathematica
Time used: 0.112 (sec). Leaf size: 124
DSolve[a*x^n*f[y'[x]] - y[x] + x*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{y(x)=a x^n f(K[1])+x K[1],x=\left (n f(K[1])^{\frac {1}{n}-1} \int _1^{K[1]}-\frac {f(K[2])^{\frac {n-1}{n}-1}}{a n}dK[2]-f(K[1])^{\frac {1}{n}-1} \int _1^{K[1]}-\frac {f(K[2])^{\frac {n-1}{n}-1}}{a n}dK[2]+c_1 f(K[1])^{\frac {1}{n}-1}\right ){}^{\frac {1}{n-1}}\right \},\{y(x),K[1]\}\right ] \]