4.3.36 \(y'(x)=x^{m-1} y(x)^{1-n} f\left (a x^m+b y(x)^n\right )\)

ODE
\[ y'(x)=x^{m-1} y(x)^{1-n} f\left (a x^m+b y(x)^n\right ) \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 221.018 (sec), leaf count = 191

\[\text {Solve}\left [c_1=\int _1^{y(x)} -\frac {K[2] \left (b n f\left (b K[2]^n+a x^m\right )+a m\right ) \left (\int _1^x \frac {a^2 b m^2 n K[1]^{m-1} K[2]^{n-1} f'\left (a K[1]^m+b K[2]^n\right )}{\left (b n f\left (a K[1]^m+b K[2]^n\right )+a m\right )^2} \, dK[1]\right )+a m K[2]^n}{K[2] \left (b n f\left (b K[2]^n+a x^m\right )+a m\right )} \, dK[2]+\int _1^x \frac {a m K[1]^{m-1} f\left (a K[1]^m+b y(x)^n\right )}{b n f\left (a K[1]^m+b y(x)^n\right )+a m} \, dK[1],y(x)\right ]\]

Maple
cpu = 0.544 (sec), leaf count = 113

\[ \left \{ {\frac {1}{m} \left (\int ^{{\frac {a{x}^{m}}{b}}+ \left (y \relax (x ) \right ) ^{n}}\! \left (\left (\sqrt [n]{{\frac {{\it \_a}\,b-am}{b}}} \right ) ^{-n}n \left (\sqrt [m]{m} \right ) ^{m} \left (-{\it \_a}\,b+am \right ) f \left (a \left (\sqrt [m]{m} \right ) ^{m}+b \left (\sqrt [n]{{\frac {{\it \_a}\,b-am}{b}}} \right ) ^{n} \right ) -a{m}^{2} \right ) ^{-1}{d{\it \_a}}b{m}^{2}-{\it \_C1}\,m+{x}^{m} \right ) }=0 \right \} \] Mathematica raw input

DSolve[y'[x] == x^(-1 + m)*f[a*x^m + b*y[x]^n]*y[x]^(1 - n),y[x],x]

Mathematica raw output

Solve[C[1] == Integrate[(a*m*f[a*K[1]^m + b*y[x]^n]*K[1]^(-1 + m))/(a*m + b*n*f[
a*K[1]^m + b*y[x]^n]), {K[1], 1, x}] + Integrate[-(((a*m + b*n*f[a*x^m + b*K[2]^
n])*Integrate[(a^2*b*m^2*n*K[1]^(-1 + m)*K[2]^(-1 + n)*Derivative[1][f][a*K[1]^m
 + b*K[2]^n])/(a*m + b*n*f[a*K[1]^m + b*K[2]^n])^2, {K[1], 1, x}]*K[2] + a*m*K[2
]^n)/((a*m + b*n*f[a*x^m + b*K[2]^n])*K[2])), {K[2], 1, y[x]}], y[x]]

Maple raw input

dsolve(diff(y(x),x) = x^(m-1)*y(x)^(1-n)*f(a*x^m+b*y(x)^n), y(x),'implicit')

Maple raw output

(Intat(1/((((_a*b-a*m)/b)^(1/n))^(-n)*n*(m^(1/m))^m*(-_a*b+a*m)*f(a*(m^(1/m))^m+
b*(((_a*b-a*m)/b)^(1/n))^n)-a*m^2),_a = 1/b*a*x^m+y(x)^n)*b*m^2-_C1*m+x^m)/m = 0