60.1.54 problem 54

Internal problem ID [10068]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 54
Date solved : Tuesday, January 28, 2025 at 04:24:37 PM
CAS classification : [_Chini, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }-a^{n} f \left (x \right )^{1-n} g^{\prime }\left (x \right ) y^{n}-\frac {f^{\prime }\left (x \right ) y}{f \left (x \right )}-f \left (x \right ) g^{\prime }\left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.226 (sec). Leaf size: 38

dsolve(diff(y(x),x) - a^n*f(x)^(1-n)*diff(g(x),x)*y(x)^n - diff(f(x),x)*y(x)/f(x) - f(x)*diff(g(x),x)=0,y(x), singsol=all)
 

\[ \frac {a y \operatorname {LerchPhi}\left (-\left (\frac {a y}{f}\right )^{n}, 1, \frac {1}{n}\right )}{n f}-a g \left (x \right )+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.338 (sec). Leaf size: 86

DSolve[D[y[x],x] - a^n*f[x]^(1-n)*D[ g[x],x]*y[x]^n - D[ f[x],x]*y[x]/f[x] - f[x]*D[ g[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [y(x) \left (a^n f(x)^{-n}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\left (\left (a^n f(x)^{-n}\right )^{\frac {1}{n}} y(x)\right )^n\right )=\int _1^xf(K[1]) \left (a^n f(K[1])^{-n}\right )^{\frac {1}{n}} g''(K[1])dK[1]+c_1,y(x)\right ] \]