4.6.20 \(x^2 y'(x)=a+b x^n+c x^2 y(x)^2\)

ODE
\[ x^2 y'(x)=a+b x^n+c x^2 y(x)^2 \] ODE Classification

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.16589 (sec), leaf count = 1577

\[\left \{\left \{y(x)\to \frac {-b^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+1} J_{\frac {\sqrt {(1-4 a c) n^2}}{n^2}-1}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}}+b^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+1} J_{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+1}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}}-b^{\frac {i \sqrt {4 a c-1}}{n}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} J_{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}}-i b^{\frac {i \sqrt {4 a c-1}}{n}} \sqrt {4 a c-1} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}+1} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} J_{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}}+b^{\frac {i \sqrt {4 a c-1}}{n}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}} \sqrt {(1-4 a c) n^2} \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}+\frac {1}{2}} J_{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {1-4 a c}}{n}\right ) c^{\frac {i \sqrt {4 a c-1}}{n}}-b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}} \left (-i \sqrt {4 a c-1} n+n+\sqrt {(1-4 a c) n^2}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}} J_{-\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{n}\right ) c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}-b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+1} J_{-\frac {\sqrt {(1-4 a c) n^2}}{n^2}-1}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{n}\right ) c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}}+b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}+1} \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+1} J_{1-\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{n}\right ) c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}+\frac {1}{2}}}{2 c n x \sqrt {x^n} \left (b^{\frac {i \sqrt {4 a c-1}}{n}} c^{\frac {i \sqrt {4 a c-1}}{n}} n^{\frac {2 \sqrt {(1-4 a c) n^2}}{n^2}} J_{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) \Gamma \left (\frac {n+\sqrt {1-4 a c}}{n}\right ) \left (x^n\right )^{\frac {i \sqrt {4 a c-1}}{n}}+b^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}} c^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}} n^{\frac {2 i \sqrt {4 a c-1}}{n}} J_{-\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\left (\frac {2 \sqrt {b} \sqrt {c} \sqrt {x^n}}{n}\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{n}\right ) \left (x^n\right )^{\frac {\sqrt {(1-4 a c) n^2}}{n^2}}\right )}\right \}\right \}\]

Maple
cpu = 0.1 (sec), leaf count = 220

\[ \left \{ y \relax (x ) ={\frac {1}{2\,cx} \left (2\,\sqrt {cb} \left ({{\sl Y}_{{\frac {\sqrt {-4\,ca+1}+n}{n}}}\left (2\,{\frac {\sqrt {cb}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {\sqrt {-4\,ca+1}+n}{n}}}\left (2\,{\frac {\sqrt {cb}{x}^{n/2}}{n}}\right )} \right ) {x}^{n/2}- \left (\sqrt {-4\,ca+1}+1 \right ) \left ({{\sl Y}_{{\frac {1}{n}\sqrt {-4\,ca+1}}}\left (2\,{\frac {\sqrt {cb}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{n}\sqrt {-4\,ca+1}}}\left (2\,{\frac {\sqrt {cb}{x}^{n/2}}{n}}\right )} \right ) \right ) \left ({{\sl Y}_{{\frac {1}{n}\sqrt {-4\,ca+1}}}\left (2\,{\frac {\sqrt {cb}{x}^{n/2}}{n}}\right )}{\it \_C1}+{{\sl J}_{{\frac {1}{n}\sqrt {-4\,ca+1}}}\left (2\,{\frac {\sqrt {cb}{x}^{n/2}}{n}}\right )} \right ) ^{-1}} \right \} \] Mathematica raw input

DSolve[x^2*y'[x] == a + b*x^n + c*x^2*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> (-(b^(Sqrt[(1 - 4*a*c)*n^2]/n^2)*c^(Sqrt[(1 - 4*a*c)*n^2]/n^2)*n^(((2*
I)*Sqrt[-1 + 4*a*c])/n)*(n - I*Sqrt[-1 + 4*a*c]*n + Sqrt[(1 - 4*a*c)*n^2])*(x^n)
^(1/2 + Sqrt[(1 - 4*a*c)*n^2]/n^2)*BesselJ[-(Sqrt[(1 - 4*a*c)*n^2]/n^2), (2*Sqrt
[b]*Sqrt[c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[1 - 4*a*c]/n]) - b^(1/2 + Sqrt[(1 
- 4*a*c)*n^2]/n^2)*c^(1/2 + Sqrt[(1 - 4*a*c)*n^2]/n^2)*n^(1 + ((2*I)*Sqrt[-1 + 4
*a*c])/n)*(x^n)^(1 + Sqrt[(1 - 4*a*c)*n^2]/n^2)*BesselJ[-1 - Sqrt[(1 - 4*a*c)*n^
2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[1 - 4*a*c]/n] + b^(
1/2 + Sqrt[(1 - 4*a*c)*n^2]/n^2)*c^(1/2 + Sqrt[(1 - 4*a*c)*n^2]/n^2)*n^(1 + ((2*
I)*Sqrt[-1 + 4*a*c])/n)*(x^n)^(1 + Sqrt[(1 - 4*a*c)*n^2]/n^2)*BesselJ[1 - Sqrt[(
1 - 4*a*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*C[1]*Gamma[1 - Sqrt[1 - 4*
a*c]/n] - b^((I*Sqrt[-1 + 4*a*c])/n)*c^((I*Sqrt[-1 + 4*a*c])/n)*n^(1 + (2*Sqrt[(
1 - 4*a*c)*n^2])/n^2)*(x^n)^(1/2 + (I*Sqrt[-1 + 4*a*c])/n)*BesselJ[Sqrt[(1 - 4*a
*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[1 - 4*a*c] + n)/n] - 
I*b^((I*Sqrt[-1 + 4*a*c])/n)*c^((I*Sqrt[-1 + 4*a*c])/n)*Sqrt[-1 + 4*a*c]*n^(1 + 
(2*Sqrt[(1 - 4*a*c)*n^2])/n^2)*(x^n)^(1/2 + (I*Sqrt[-1 + 4*a*c])/n)*BesselJ[Sqrt
[(1 - 4*a*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[1 - 4*a*c] +
 n)/n] + b^((I*Sqrt[-1 + 4*a*c])/n)*c^((I*Sqrt[-1 + 4*a*c])/n)*n^((2*Sqrt[(1 - 4
*a*c)*n^2])/n^2)*Sqrt[(1 - 4*a*c)*n^2]*(x^n)^(1/2 + (I*Sqrt[-1 + 4*a*c])/n)*Bess
elJ[Sqrt[(1 - 4*a*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[1 - 
4*a*c] + n)/n] - b^(1/2 + (I*Sqrt[-1 + 4*a*c])/n)*c^(1/2 + (I*Sqrt[-1 + 4*a*c])/
n)*n^(1 + (2*Sqrt[(1 - 4*a*c)*n^2])/n^2)*(x^n)^(1 + (I*Sqrt[-1 + 4*a*c])/n)*Bess
elJ[-1 + Sqrt[(1 - 4*a*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt
[1 - 4*a*c] + n)/n] + b^(1/2 + (I*Sqrt[-1 + 4*a*c])/n)*c^(1/2 + (I*Sqrt[-1 + 4*a
*c])/n)*n^(1 + (2*Sqrt[(1 - 4*a*c)*n^2])/n^2)*(x^n)^(1 + (I*Sqrt[-1 + 4*a*c])/n)
*BesselJ[1 + Sqrt[(1 - 4*a*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(
Sqrt[1 - 4*a*c] + n)/n])/(2*c*n*x*Sqrt[x^n]*(b^(Sqrt[(1 - 4*a*c)*n^2]/n^2)*c^(Sq
rt[(1 - 4*a*c)*n^2]/n^2)*n^(((2*I)*Sqrt[-1 + 4*a*c])/n)*(x^n)^(Sqrt[(1 - 4*a*c)*
n^2]/n^2)*BesselJ[-(Sqrt[(1 - 4*a*c)*n^2]/n^2), (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]
*C[1]*Gamma[1 - Sqrt[1 - 4*a*c]/n] + b^((I*Sqrt[-1 + 4*a*c])/n)*c^((I*Sqrt[-1 + 
4*a*c])/n)*n^((2*Sqrt[(1 - 4*a*c)*n^2])/n^2)*(x^n)^((I*Sqrt[-1 + 4*a*c])/n)*Bess
elJ[Sqrt[(1 - 4*a*c)*n^2]/n^2, (2*Sqrt[b]*Sqrt[c]*Sqrt[x^n])/n]*Gamma[(Sqrt[1 - 
4*a*c] + n)/n]))}}

Maple raw input

dsolve(x^2*diff(y(x),x) = a+b*x^n+c*x^2*y(x)^2, y(x),'implicit')

Maple raw output

y(x) = 1/2*(2*(c*b)^(1/2)*(BesselY(((-4*a*c+1)^(1/2)+n)/n,2*(c*b)^(1/2)*x^(1/2*n
)/n)*_C1+BesselJ(((-4*a*c+1)^(1/2)+n)/n,2*(c*b)^(1/2)*x^(1/2*n)/n))*x^(1/2*n)-((
-4*a*c+1)^(1/2)+1)*(BesselY((-4*a*c+1)^(1/2)/n,2*(c*b)^(1/2)*x^(1/2*n)/n)*_C1+Be
sselJ((-4*a*c+1)^(1/2)/n,2*(c*b)^(1/2)*x^(1/2*n)/n)))/x/c/(BesselY((-4*a*c+1)^(1
/2)/n,2*(c*b)^(1/2)*x^(1/2*n)/n)*_C1+BesselJ((-4*a*c+1)^(1/2)/n,2*(c*b)^(1/2)*x^
(1/2*n)/n))