Internal problem ID [8132]
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: 551.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{n}-f \relax (x ) g \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.03 (sec). Leaf size: 43
dsolve(diff(y(x),x)^n-f(x)*g(y(x))=0,y(x), singsol=all)
\[ \int _{}^{y \relax (x )}g \left (\textit {\_a} \right )^{-\frac {1}{n}}d \textit {\_a} +\int _{}^{x}-\left (f \left (\textit {\_a} \right ) g \left (y \relax (x )\right )\right )^{\frac {1}{n}} g \left (y \relax (x )\right )^{-\frac {1}{n}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.264 (sec). Leaf size: 41
DSolve[-(f[x]*g[y[x]]) + y'[x]^n==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} 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 ] \\ \end{align*}