4.2.46 $y^{\prime }(x)=f(x)y(x)+g(x)y(x{)}^k$

ODE
$$y^{\prime }(x)=f(x)y(x)+g(x)y(x{)}^k$$ ODE Classification

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica ✓
cpu = 0.493421 (sec), leaf count = 64

$$\left\{\{y(x)\to \left(e^{-(k-1){\int }_1^{x}f(K[1])dK[1]}(c_1-(k-1){\int }_1^{x}g(K[2])e^{(k-1){\int }_1^{K[2]}f(K[1])dK[1]}dK[2])\right){}^{\frac{1}{1-k}}\}\right\}$$

Maple ✓
cpu = 0.019 (sec), leaf count = 48

$$\{{\left(y\left(x\right)\right)}^{1-k}+\frac{-\int -{\mathrm{e}}^{\int f\left(x\right)(k-1)\mathrm{d}x}g\left(x\right)(k-1)\mathrm{d}x-\mathit{\_}\mathit{C}\mathit{1}}{{\mathrm{e}}^{\int f\left(x\right)(k-1)\mathrm{d}x}}=0\}$$ 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)*Integrate[E^((-1 + k)*Integrate[f[K[1]], {K[1], 1, K
[2]}])*g[K[2]], {K[2], 1, x}])/E^((-1 + k)*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),'implicit')

Maple raw output

y(x)^(1-k)+(-Int(-exp(Int(f(x)*(k-1),x))*g(x)*(k-1),x)-_C1)/exp(Int(f(x)*(k-1),x
)) = 0