Internal problem ID [7634]
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)]]]
Solve \begin {gather*} \boxed {y^{\prime }-f \relax (x )^{1-n} g^{\prime }\relax (x ) y^{n} \left (a g \relax (x )+b \right )^{-n}-\frac {f^{\prime }\relax (x ) y}{f \relax (x )}-f \relax (x ) g^{\prime }\relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.04 (sec). Leaf size: 214
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)
\[ y \relax (x ) = \frac {\RootOf \left (-f \relax (x )^{n} \left (a g \relax (x )+b \right )^{n} \left (\left (a g \relax (x )+b \right )^{-n -1} n a f \relax (x )^{-n +2} \left (\frac {d}{d x}g \relax (x )\right )^{3}\right )^{n} \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \left (a g \relax (x )+b \right )^{n} f \relax (x )^{n} \left (\left (a g \relax (x )+b \right )^{-n -1} n a f \relax (x )^{-n +2} \left (\frac {d}{d x}g \relax (x )\right )^{3}\right )^{n}-f \relax (x )^{n} \left (a g \relax (x )+b \right )^{n} \left (\left (a g \relax (x )+b \right )^{-n -1} n a f \relax (x )^{-n +2} \left (\frac {d}{d x}g \relax (x )\right )^{3}\right )^{n}-\textit {\_a}^{n} \left (\left (a g \relax (x )+b \right )^{-n} f \relax (x )^{-n +1} \left (\frac {d}{d x}g \relax (x )\right )\right )^{n} \left (\left (\frac {d}{d x}g \relax (x )\right ) f \relax (x )\right )^{2 n} n^{n}}d \textit {\_a} \right )-\ln \left (a g \relax (x )+b \right )+c_{1}\right ) \left (a g \relax (x )+b \right ) f \relax (x )}{a} \]
✓ Solution by Mathematica
Time used: 0.391 (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]
\[ \operatorname {Solve}\left [\int _1^{\left (f(x)^{-n} (b+a g(x))^{-n}\right )^{\frac {1}{n}} y(x)}\frac {1}{K[1]^n-\left (a^n\right )^{\frac {1}{n}} K[1]+1}dK[1]=\frac {f(x) (a g(x)+b) \log (a g(x)+b) \left (f(x)^{-n} (a g(x)+b)^{-n}\right )^{\frac {1}{n}}}{a}+c_1,y(x)\right ] \]