Internal problem ID [9582]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, First-Order differential equations
Problem number: 1.1.5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {g \left (x \right ) y^{\prime }-f_{1} \left (x \right ) y-f_{n} \left (x \right ) y^{n}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 119
dsolve(g(x)*diff(y(x),x)=f__1(x)*y(x)+f__n(x)*y(x)^n,y(x), singsol=all)
\[ y \left (x \right ) = {\left (\int \left (-\frac {n \,{\mathrm e}^{\int \left (\frac {f_{1} \left (x \right ) n}{g \left (x \right )}-\frac {f_{1} \left (x \right )}{g \left (x \right )}\right )d x} f_{n} \left (x \right )}{g \left (x \right )}+\frac {{\mathrm e}^{\int \left (\frac {f_{1} \left (x \right ) n}{g \left (x \right )}-\frac {f_{1} \left (x \right )}{g \left (x \right )}\right )d x} f_{n} \left (x \right )}{g \left (x \right )}\right )d x +c_{1} \right )}^{-\frac {1}{n -1}} {\mathrm e}^{\frac {\left (\int \frac {f_{1} \left (x \right )}{g \left (x \right )}d x \right ) n}{n -1}} {\mathrm e}^{\int -\frac {f_{1} \left (x \right )}{\left (n -1\right ) g \left (x \right )}d x} \]
✓ Solution by Mathematica
Time used: 10.848 (sec). Leaf size: 84
DSolve[g[x]*y'[x]==f1[x]*y[x]+fn[x]*y[x]^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\exp \left (-\left ((n-1) \int _1^x\frac {\text {f1}(K[1])}{g(K[1])}dK[1]\right )\right ) \left (-(n-1) \int _1^x\frac {\exp \left ((n-1) \int _1^{K[2]}\frac {\text {f1}(K[1])}{g(K[1])}dK[1]\right ) \text {fn}(K[2])}{g(K[2])}dK[2]+c_1\right )\right ){}^{\frac {1}{1-n}} \\ \end{align*}