4.15.24 $f(x)y(x{)}^m{y}^{\prime }(x)+g(x)y(x{)}^{m+1}+h(x)y(x{)}^n=0$

ODE
$$f(x)y(x{)}^m{y}^{\prime }(x)+g(x)y(x{)}^{m+1}+h(x)y(x{)}^n=0$$ ODE Classification

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica ✓
cpu = 1.04416 (sec), leaf count = 93

$$\left\{\{y(x)\to (\exp ((m-n+1){\int }_1^x-\frac{g(K[1])}{f(K[1])}dK[1])((m-n+1){\int }_1^x-\frac{h(K[2])\exp (-(m-n+1){\int }_1^{K[2]}-\frac{g(K[1])}{f(K[1])}dK[1])}{f(K[2])}dK[2]+c_1)){}^{\frac{1}{m-n+1}}\}\right\}$$

Maple ✓
cpu = 0.035 (sec), leaf count = 70

$$\{{\left(y\left(x\right)\right)}^{m-n+1}+1(-\int -\frac{h\left(x\right)(m-n+1)}{f\left(x\right)}{\mathrm{e}}^{\int \frac{g\left(x\right)(m-n+1)}{f\left(x\right)}\mathrm{d}x}\mathrm{d}x-\mathit{\_}\mathit{C}\mathit{1}){\left({\mathrm{e}}^{\int \frac{g\left(x\right)(m-n+1)}{f\left(x\right)}\mathrm{d}x}\right)}^{-1}=0\}$$ Mathematica raw input

DSolve[g[x]*y[x]^(1 + m) + h[x]*y[x]^n + f[x]*y[x]^m*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (E^((1 + m - n)*Integrate[-(g[K[1]]/f[K[1]]), {K[1], 1, x}])*(C[1] + (
1 + m - n)*Integrate[-(h[K[2]]/(E^((1 + m - n)*Integrate[-(g[K[1]]/f[K[1]]), {K[
1], 1, K[2]}])*f[K[2]])), {K[2], 1, x}]))^(1 + m - n)^(-1)}}

Maple raw input

dsolve(f(x)*y(x)^m*diff(y(x),x)+g(x)*y(x)^(m+1)+h(x)*y(x)^n = 0, y(x),'implicit')

Maple raw output

y(x)^(m-n+1)+(-Int(-exp(Int(1/f(x)*g(x)*(m-n+1),x))*h(x)/f(x)*(m-n+1),x)-_C1)/ex
p(Int(1/f(x)*g(x)*(m-n+1),x)) = 0