Internal problem ID [3354]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 4
Problem number: 95.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime }-f \left (x \right ) y-g \left (x \right ) y^{k}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 51
dsolve(diff(y(x),x) = f(x)*y(x)+g(x)*y(x)^k,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\int f \left (x \right )d x} {\left (-k \left (\int g \left (x \right ) {\mathrm e}^{\left (k -1\right ) \left (\int f \left (x \right )d x \right )}d x \right )+c_{1} +\int g \left (x \right ) {\mathrm e}^{\left (k -1\right ) \left (\int f \left (x \right )d x \right )}d x \right )}^{-\frac {1}{k -1}} \]
✓ Solution by Mathematica
Time used: 11.421 (sec). Leaf size: 129
DSolve[y'[x]==f[x] y[x]+g[x]y[x]^k,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\exp \left (-\left ((k-1) \int _1^xf(K[1])dK[1]\right )\right ) \left (-(k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]+c_1\right )\right ){}^{\frac {1}{1-k}} \\ y(x)\to \left ((k-1) \left (-\exp \left (-\left ((k-1) \int _1^xf(K[1])dK[1]\right )\right )\right ) \int _1^x\exp \left ((k-1) \int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]\right ){}^{\frac {1}{1-k}} \\ \end{align*}