ODE
\[ y'(x)=f(x) y(x)+g(x) y(x)^k \] ODE Classification
[_Bernoulli]
Book solution method
The Bernoulli ODE
Mathematica ✓
cpu = 0.351682 (sec), leaf count = 67
\[\left \{\left \{y(x)\to \left (\exp \left (-\left ((k-1) \int _1^xf(K[1])dK[1]\right )\right ) \left (-(k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]+c_1\right )\right ){}^{\frac {1}{1-k}}\right \}\right \}\]
Maple ✓
cpu = 0.117 (sec), leaf count = 81
\[\left [y \left (x \right ) = \left (\int \left (-k \,{\mathrm e}^{\int \left (f \left (x \right ) k -f \left (x \right )\right )d x} g \left (x \right )+{\mathrm e}^{\int \left (f \left (x \right ) k -f \left (x \right )\right )d x} g \left (x \right )\right )d x +\textit {\_C1} \right )^{-\frac {1}{k -1}} {\mathrm e}^{\frac {\left (\int f \left (x \right )d x \right ) k}{k -1}} {\mathrm e}^{\int -\frac {f \left (x \right )}{k -1}d x}\right ]\] Mathematica raw input
DSolve[y'[x] == f[x]*y[x] + g[x]*y[x]^k,y[x],x]
Mathematica raw output
{{y[x] -> ((C[1] - (-1 + k)*Inactive[Integrate][E^((-1 + k)*Inactive[Integrate][f[K[1]], {K[1], 1, K[2]}])*g[K[2]], {K[2], 1, x}])/E^((-1 + k)*Inactive[Integrate][f[K[1]], {K[1], 1, x}]))^(1 - k)^(-1)}}
Maple raw input
dsolve(diff(y(x),x) = f(x)*y(x)+g(x)*y(x)^k, y(x))
Maple raw output
[y(x) = 1/((-k*Int(exp(Int(f(x),x)*k)/exp(Int(f(x),x))*g(x),x)+_C1+Int(exp(Int(f(x),x)*k)/exp(Int(f(x),x))*g(x),x))^(1/(k-1)))*exp(1/(k-1)*Int(f(x),x)*k)/exp(1/(k-1)*Int(f(x),x))]