4.23.11 \(y'(x)^n=a x^r+b y'(x)^s\)

ODE
\[ y'(x)^n=a x^r+b y'(x)^s \] ODE Classification

[_quadrature]

Book solution method
Change of variable

Mathematica ✓
cpu = 2.17155 (sec), leaf count = 249

\[\text {Solve}\left [\left \{y(x)=\frac {K[1]^{1-s} \left (1-\frac {K[1]^{n-s}}{b}\right )^{-1/r} \left (\frac {K[1]^n-b K[1]^s}{a}\right )^{\frac {1}{r}} \left (n (r+s) K[1]^n \, _2F_1\left (\frac {r-1}{r},\frac {n r-s r+r+s}{n r-r s};\frac {2 n r-2 s r+r+s}{n r-r s};\frac {K[1]^{n-s}}{b}\right )+b s (r (-n+s-1)-s) K[1]^s \, _2F_1\left (\frac {r-1}{r},\frac {r+s}{n r-r s};\frac {r+s}{n r-r s}+1;\frac {K[1]^{n-s}}{b}\right )\right )}{b (r+s) (r (-n+s-1)-s)}+c_1,\left (\frac {K[1]^n-b K[1]^s}{a}\right )^{\frac {1}{r}}=x\right \},\{y(x),K[1]\}\right ]\]

Maple ✓
cpu = 0.994 (sec), leaf count = 25

\[[y \left (x \right ) = \int \RootOf \left (-\textit {\_Z}^{n}+a \,x^{r}+b \,\textit {\_Z}^{s}\right )d x +\textit {\_C1}]\] Mathematica raw input

DSolve[y'[x]^n == a*x^r + b*y'[x]^s,y[x],x]

Mathematica raw output

Solve[{y[x] == C[1] + (K[1]^(1 - s)*((K[1]^n - b*K[1]^s)/a)^r^(-1)*(n*(r + s)*Hy
pergeometric2F1[(-1 + r)/r, (r + n*r + s - r*s)/(n*r - r*s), (r + 2*n*r + s - 2*
r*s)/(n*r - r*s), K[1]^(n - s)/b]*K[1]^n + b*s*(-s + r*(-1 - n + s))*Hypergeomet
ric2F1[(-1 + r)/r, (r + s)/(n*r - r*s), 1 + (r + s)/(n*r - r*s), K[1]^(n - s)/b]
*K[1]^s))/(b*(r + s)*(-s + r*(-1 - n + s))*(1 - K[1]^(n - s)/b)^r^(-1)), ((K[1]^
n - b*K[1]^s)/a)^r^(-1) == x}, {y[x], K[1]}]

Maple raw input

dsolve(diff(y(x),x)^n = a*x^r+b*diff(y(x),x)^s, y(x))

Maple raw output

[y(x) = Int(RootOf(-_Z^n+a*x^r+b*_Z^s),x)+_C1]