Internal problem ID [8390]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 53.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Chini, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {y^{\prime }-f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n} \left (a g \left (x \right )+b \right )^{-n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}=g^{\prime }\left (x \right ) f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 277
dsolve(diff(y(x),x) - f(x)^(1-n)*diff(g(x),x)*y(x)^n/(a*g(x)+b)^n - diff(f(x),x)*y(x)/f(x) - f(x)*diff(g(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {\left (f \left (x \right )^{-n +1} \left (a g \left (x \right )+b \right )^{-n} \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-1-n} \left (f \left (x \right ) \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-2 n +1} \left (a n f \left (x \right )^{-n +2} \left (a g \left (x \right )+b \right )^{-1-n} \left (\frac {d}{d x}g \left (x \right )\right )^{3}\right )^{n} n^{-n}}{-\textit {\_a} \left (f \left (x \right )^{-n +1} \left (a g \left (x \right )+b \right )^{-n} \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-1-n} \left (f \left (x \right ) \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-2 n +1} \left (a n f \left (x \right )^{-n +2} \left (a g \left (x \right )+b \right )^{-1-n} \left (\frac {d}{d x}g \left (x \right )\right )^{3}\right )^{n} n^{-n}+\left (f \left (x \right )^{-n +1} \left (a g \left (x \right )+b \right )^{-n} \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-1-n} \left (f \left (x \right ) \left (\frac {d}{d x}g \left (x \right )\right )\right )^{-2 n +1} \left (a n f \left (x \right )^{-n +2} \left (a g \left (x \right )+b \right )^{-1-n} \left (\frac {d}{d x}g \left (x \right )\right )^{3}\right )^{n} n^{-n}+\textit {\_a}^{n}}d \textit {\_a} -\ln \left (a g \left (x \right )+b \right )+c_{1} \right ) \left (a g \left (x \right )+b \right ) f \left (x \right )}{a} \]
✓ Solution by Mathematica
Time used: 0.405 (sec). Leaf size: 96
DSolve[y'[x] - f[x]^(1-n)*g'[x]*y[x]^n/(a*g[x]+b)^n - f'[x]*y[x]/f[x] - f[x]*g'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{\left (f(x)^{-n} (b+a g(x))^{-n}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (a^n\right )^{\frac {1}{n}} K[1]+1}dK[1]=\frac {f(x) (a g(x)+b) \log (a g(x)+b) \left (f(x)^{-n} (a g(x)+b)^{-n}\right )^{\frac {1}{n}}}{a}+c_1,y(x)\right ] \]