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

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

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.473038 (sec), leaf count = 96

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

Maple
cpu = 0.186 (sec), leaf count = 221

$\left [y \left (x \right ) = \left (-m \left (\int \frac {{\mathrm e}^{\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right ) m} {\mathrm e}^{\int -\frac {g \left (x \right ) n}{f \left (x \right )}d x} {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x \right )+n \left (\int \frac {{\mathrm e}^{\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right ) m} {\mathrm e}^{\int -\frac {g \left (x \right ) n}{f \left (x \right )}d x} {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x \right )+\textit {\_C1} -\left (\int \frac {{\mathrm e}^{\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right ) m} {\mathrm e}^{\int -\frac {g \left (x \right ) n}{f \left (x \right )}d x} {\mathrm e}^{\int \frac {g \left (x \right )}{f \left (x \right )}d x} h \left (x \right )}{f \left (x \right )}d x \right )\right )^{\frac {1}{m -n +1}} {\mathrm e}^{\int -\frac {m g \left (x \right )}{\left (m -n +1\right ) f \left (x \right )}d x} {\mathrm e}^{\frac {\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right ) n}{m -n +1}} {\mathrm e}^{\int -\frac {g \left (x \right )}{\left (m -n +1\right ) f \left (x \right )}d x}\right ]$ 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)*Inactive[Integrate][-(g[K[1]]/f[K[1]]), {K[1], 1, x}])
*(C[1] + (1 + m - n)*Inactive[Integrate][-(h[K[2]]/(E^((1 + m - n)*Inactive[Inte
grate][-(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))

Maple raw output

[y(x) = (-m*Int(exp(Int(1/f(x)*g(x),x)*m)/exp(Int(1/f(x)*g(x),x)*n)*exp(Int(1/f(
x)*g(x),x))*h(x)/f(x),x)+n*Int(exp(Int(1/f(x)*g(x),x)*m)/exp(Int(1/f(x)*g(x),x)*
n)*exp(Int(1/f(x)*g(x),x))*h(x)/f(x),x)+_C1-Int(exp(Int(1/f(x)*g(x),x)*m)/exp(In
t(1/f(x)*g(x),x)*n)*exp(Int(1/f(x)*g(x),x))*h(x)/f(x),x))^(1/(m-n+1))/exp(1/(m-n
+1)*Int(1/f(x)*g(x),x)*m)*exp(1/(m-n+1)*Int(1/f(x)*g(x),x)*n)/exp(1/(m-n+1)*Int(
1/f(x)*g(x),x))]