4.12.43 \(x^7 y(x) y'(x)=5 x^3 y(x)+2 \left (x^2+1\right )\)

ODE
\[ x^7 y(x) y'(x)=5 x^3 y(x)+2 \left (x^2+1\right ) \] ODE Classification

[_rational, [_Abel, `2nd type``class B`]]

Book solution method
Abel ODE, Second kind

Mathematica
cpu = 1.16006 (sec), leaf count = 98

\[\text {Solve}\left [c_1=\frac {i \left (\left (x^3 y(x)+1\right ) \sqrt [4]{x^4 y(x)^2+\frac {1}{x^2}+2 x y(x)+1} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {\left (y(x) x^3+1\right )^2}{x^2}\right )+2 x^2\right )}{2 x \sqrt [4]{-\frac {\left (x^3 y(x)+1\right )^2}{x^2}-1}},y(x)\right ]\]

Maple
cpu = 0.055 (sec), leaf count = 96

\[ \left \{ {\it \_C1}+{1 \left (-{\frac {1+{x}^{3}y \relax (x ) }{x}{\mbox {$_2$F$_1$}({\frac {1}{2}},{\frac {5}{4}};\,{\frac {3}{2}};\,-{\frac { \left (1+{x}^{3}y \relax (x ) \right ) ^{2}}{{x}^{2}}})}\sqrt [4]{{\frac {{x}^{6} \left (y \relax (x ) \right ) ^{2}+2\,{x}^{3}y \relax (x ) +{x}^{2}+1}{{x}^{2}}}}}-2\,x \right ) {\frac {1}{\sqrt [4]{{\frac {{x}^{6} \left (y \relax (x ) \right ) ^{2}+2\,{x}^{3}y \relax (x ) +{x}^{2}+1}{{x}^{2}}}}}}}=0 \right \} \] Mathematica raw input

DSolve[x^7*y[x]*y'[x] == 2*(1 + x^2) + 5*x^3*y[x],y[x],x]

Mathematica raw output

Solve[C[1] == ((I/2)*(2*x^2 + Hypergeometric2F1[1/2, 5/4, 3/2, -((1 + x^3*y[x])^
2/x^2)]*(1 + x^3*y[x])*(1 + x^(-2) + 2*x*y[x] + x^4*y[x]^2)^(1/4)))/(x*(-1 - (1 
+ x^3*y[x])^2/x^2)^(1/4)), y[x]]

Maple raw input

dsolve(x^7*y(x)*diff(y(x),x) = 2*x^2+2+5*x^3*y(x), y(x),'implicit')

Maple raw output

_C1+(-(1+x^3*y(x))/x*hypergeom([1/2, 5/4],[3/2],-(1+x^3*y(x))^2/x^2)*((x^6*y(x)^
2+2*x^3*y(x)+x^2+1)/x^2)^(1/4)-2*x)/((x^6*y(x)^2+2*x^3*y(x)+x^2+1)/x^2)^(1/4) = 
0