##### 4.2.46 $$y'(x)=f(x) y(x)+g(x) y(x)^k$$

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[Integrat
e][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))]