4.2.20 \(y'(x)=a x^m+b x^n y(x)^2\)

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

[_Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.104942 (sec), leaf count = 1067

\[\left \{\left \{y(x)\to -\frac {a^{-\frac {n+1}{2 (m+n+2)}} b^{-\frac {2 m+3 n+5}{2 (m+n+2)}} (m+n+1)^{\frac {n+1}{m+n+2}} \left ((m+n+1)^2\right )^{\frac {n+1}{m+n+2}-\frac {1}{2}} x^{-n-1} \left (x^{m+n+1}\right )^{-\frac {n+1}{2 (m+n+1)}} \left (a^{\frac {n+1}{2 (m+n+2)}} b^{\frac {n+1}{2 (m+n+2)}} (m+n+1)^{-\frac {n+1}{m+n+2}} (m+n+2) \left (-\sqrt {a} \sqrt {b} (m+n+1) J_{\frac {m+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}+\sqrt {a} \sqrt {b} (m+n+1) J_{-\frac {m+2 n+3}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}+(n+1) \sqrt {(m+n+1)^2} J_{-\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right ) c_1 \Gamma \left (\frac {m+1}{m+n+2}\right ) \left (x^{m+n+1}\right )^{\frac {n+1}{2 (m+n+1)}}+a^{\frac {n+1}{2 (m+n+2)}} b^{\frac {n+1}{2 (m+n+2)}} (n+1)^2 (m+n+1)^{\frac {n+1}{m+n+2}} \left ((m+n+1)^2\right )^{\frac {m-n}{2 (m+n+2)}} J_{\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \Gamma \left (\frac {n+1}{m+n+2}\right ) \left (x^{m+n+1}\right )^{\frac {n+1}{2 (m+n+1)}}+a^{\frac {m+2 n+3}{2 (m+n+2)}} b^{\frac {m+2 n+3}{2 (m+n+2)}} (n+1) (m+n+1)^{\frac {m+2 n+3}{m+n+2}} \left ((m+n+1)^2\right )^{-\frac {n+1}{m+n+2}} \left (J_{-\frac {m+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-J_{\frac {n+1}{m+n+2}+1}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right ) \Gamma \left (\frac {n+1}{m+n+2}\right ) \left (x^{m+n+1}\right )^{\frac {m+2 n+3}{2 (m+n+1)}}\right )}{2 \left ((m+n+2) J_{-\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) c_1 \Gamma \left (\frac {m+1}{m+n+2}\right ) \left ((m+n+1)^2\right )^{\frac {n+1}{m+n+2}}+(n+1) (m+n+1)^{\frac {2 (n+1)}{m+n+2}} J_{\frac {n+1}{m+n+2}}\left (\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \Gamma \left (\frac {n+1}{m+n+2}\right )\right )}\right \}\right \}\]

Maple
cpu = 0.213 (sec), leaf count = 179

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

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

Mathematica raw output

{{y[x] -> -((1 + m + n)^((1 + n)/(2 + m + n))*((1 + m + n)^2)^(-1/2 + (1 + n)/(2
 + m + n))*x^(-1 - n)*((a^((1 + n)/(2*(2 + m + n)))*b^((1 + n)/(2*(2 + m + n)))*
(2 + m + n)*(x^(1 + m + n))^((1 + n)/(2*(1 + m + n)))*(-(Sqrt[a]*Sqrt[b]*(1 + m 
+ n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))/2)*BesselJ[(1 + m)/(2 + m + n), (2*
Sqrt[a]*Sqrt[b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))/2))/(Sqrt[(1
 + m + n)^2]*(2 + m + n))]) + (1 + n)*Sqrt[(1 + m + n)^2]*BesselJ[-((1 + n)/(2 +
 m + n)), (2*Sqrt[a]*Sqrt[b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))
/2))/(Sqrt[(1 + m + n)^2]*(2 + m + n))] + Sqrt[a]*Sqrt[b]*(1 + m + n)*(x^(1 + m 
+ n))^((1 + (1 + m + n)^(-1))/2)*BesselJ[-((3 + m + 2*n)/(2 + m + n)), (2*Sqrt[a
]*Sqrt[b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))/2))/(Sqrt[(1 + m +
 n)^2]*(2 + m + n))])*C[1]*Gamma[(1 + m)/(2 + m + n)])/(1 + m + n)^((1 + n)/(2 +
 m + n)) + a^((1 + n)/(2*(2 + m + n)))*b^((1 + n)/(2*(2 + m + n)))*(1 + n)^2*(1 
+ m + n)^((1 + n)/(2 + m + n))*((1 + m + n)^2)^((m - n)/(2*(2 + m + n)))*(x^(1 +
 m + n))^((1 + n)/(2*(1 + m + n)))*BesselJ[(1 + n)/(2 + m + n), (2*Sqrt[a]*Sqrt[
b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))/2))/(Sqrt[(1 + m + n)^2]*
(2 + m + n))]*Gamma[(1 + n)/(2 + m + n)] + (a^((3 + m + 2*n)/(2*(2 + m + n)))*b^
((3 + m + 2*n)/(2*(2 + m + n)))*(1 + n)*(1 + m + n)^((3 + m + 2*n)/(2 + m + n))*
(x^(1 + m + n))^((3 + m + 2*n)/(2*(1 + m + n)))*(BesselJ[-((1 + m)/(2 + m + n)),
 (2*Sqrt[a]*Sqrt[b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))/2))/(Sqr
t[(1 + m + n)^2]*(2 + m + n))] - BesselJ[1 + (1 + n)/(2 + m + n), (2*Sqrt[a]*Sqr
t[b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)^(-1))/2))/(Sqrt[(1 + m + n)^2
]*(2 + m + n))])*Gamma[(1 + n)/(2 + m + n)])/((1 + m + n)^2)^((1 + n)/(2 + m + n
))))/(2*a^((1 + n)/(2*(2 + m + n)))*b^((5 + 2*m + 3*n)/(2*(2 + m + n)))*(x^(1 + 
m + n))^((1 + n)/(2*(1 + m + n)))*(((1 + m + n)^2)^((1 + n)/(2 + m + n))*(2 + m 
+ n)*BesselJ[-((1 + n)/(2 + m + n)), (2*Sqrt[a]*Sqrt[b]*(1 + m + n)*(x^(1 + m + 
n))^((1 + (1 + m + n)^(-1))/2))/(Sqrt[(1 + m + n)^2]*(2 + m + n))]*C[1]*Gamma[(1
 + m)/(2 + m + n)] + (1 + n)*(1 + m + n)^((2*(1 + n))/(2 + m + n))*BesselJ[(1 + 
n)/(2 + m + n), (2*Sqrt[a]*Sqrt[b]*(1 + m + n)*(x^(1 + m + n))^((1 + (1 + m + n)
^(-1))/2))/(Sqrt[(1 + m + n)^2]*(2 + m + n))]*Gamma[(1 + n)/(2 + m + n)]))}}

Maple raw input

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

Maple raw output

y(x) = (BesselY((m+1)/(m+n+2),2*(a*b)^(1/2)*x^(1/2*m+1/2*n+1)/(m+n+2))*_C1+Besse
lJ((m+1)/(m+n+2),2*(a*b)^(1/2)*x^(1/2*m+1/2*n+1)/(m+n+2)))*x^(1/2*m+1/2*n+1)*(a*
b)^(1/2)/(BesselY((-n-1)/(m+n+2),2*(a*b)^(1/2)*x^(1/2*m+1/2*n+1)/(m+n+2))*_C1+Be
sselJ((-n-1)/(m+n+2),2*(a*b)^(1/2)*x^(1/2*m+1/2*n+1)/(m+n+2)))*x^(-n)/b/x