Internal problem ID [8887]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 552.
ODE order: 1.
ODE degree: 0.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{n}-f \left (x \right ) g \left (y\right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 43
dsolve(diff(y(x),x)^n-f(x)*g(y(x))=0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}g \left (\textit {\_a} \right )^{-\frac {1}{n}}d \textit {\_a} -g \left (y \left (x \right )\right )^{-\frac {1}{n}} \left (\int _{}^{x}\left (f \left (\textit {\_a} \right ) g \left (y \left (x \right )\right )\right )^{\frac {1}{n}}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.239 (sec). Leaf size: 41
DSolve[-(f[x]*g[y[x]]) + y'[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}g(K[1])^{-1/n}dK[1]\&\right ]\left [\int _1^xf(K[2])^{\frac {1}{n}}dK[2]+c_1\right ] \]