2.53 problem 53

Internal problem ID [9640]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 53.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_rational, _Riccati]

Solve \begin {gather*} \boxed {x^{2} y^{\prime }-y^{2} c \,x^{2}-\left (a \,x^{n}+b \right ) x y-\alpha \,x^{2 n}-\beta \,x^{n}-\gamma =0} \end {gather*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 1036

dsolve(x^2*diff(y(x),x)=c*x^2*y(x)^2+(a*x^n+b)*x*y(x)+alpha*x^(2*n)+beta*x^n+gamma,y(x), singsol=all)
 

\[ y \relax (x ) = -\frac {\left (x^{n} \sqrt {a^{2}-4 \alpha c}\, c_{1} a^{3}-4 x^{n} \sqrt {a^{2}-4 \alpha c}\, c_{1} a \alpha c +x^{n} c_{1} a^{4}-8 x^{n} c_{1} a^{2} \alpha c +16 x^{n} c_{1} \alpha ^{2} c^{2}+\sqrt {a^{2}-4 \alpha c}\, c_{1} a^{2} b -\sqrt {a^{2}-4 \alpha c}\, c_{1} a^{2} n -4 \sqrt {a^{2}-4 \alpha c}\, c_{1} \alpha b c +4 \sqrt {a^{2}-4 \alpha c}\, c_{1} \alpha c n +c_{1} a^{3} b -c_{1} a^{3} n -2 c_{1} a^{2} \beta c -4 c_{1} a \alpha b c +4 c_{1} a \alpha c n +8 c_{1} \alpha \beta \,c^{2}+\sqrt {a^{2}-4 \alpha c}\, c_{1} a^{2}-4 \sqrt {a^{2}-4 \alpha c}\, c_{1} \alpha c +c_{1} a^{3}-4 c_{1} a \alpha c \right ) \WhittakerW \left (-\frac {a b -n a -2 \beta c +a}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )+\left (-2 \sqrt {a^{2}-4 \alpha c}\, c_{1} a^{2} n +8 \sqrt {a^{2}-4 \alpha c}\, c_{1} \alpha c n \right ) \WhittakerW \left (\frac {2 \sqrt {a^{2}-4 \alpha c}\, n -a b +n a +2 \beta c -a}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )+\left (x^{n} \sqrt {a^{2}-4 \alpha c}\, a^{3}-4 x^{n} \sqrt {a^{2}-4 \alpha c}\, a \alpha c +x^{n} a^{4}-8 x^{n} a^{2} \alpha c +16 x^{n} \alpha ^{2} c^{2}+\sqrt {a^{2}-4 \alpha c}\, a^{2} b -\sqrt {a^{2}-4 \alpha c}\, a^{2} n -4 \sqrt {a^{2}-4 \alpha c}\, \alpha b c +4 \sqrt {a^{2}-4 \alpha c}\, \alpha c n +a^{3} b -n \,a^{3}-2 a^{2} \beta c -4 \alpha b c a +4 a \alpha c n +8 \alpha \beta \,c^{2}+\sqrt {a^{2}-4 \alpha c}\, a^{2}-4 \sqrt {a^{2}-4 \alpha c}\, \alpha c +a^{3}-4 a \alpha c \right ) \WhittakerM \left (-\frac {a b -n a -2 \beta c +a}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )+\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}\, a^{2}-4 \sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}\, \alpha c +\sqrt {a^{2}-4 \alpha c}\, a^{2} n -4 \sqrt {a^{2}-4 \alpha c}\, \alpha c n -a^{3} b +n \,a^{3}+2 a^{2} \beta c +4 \alpha b c a -4 a \alpha c n -8 \alpha \beta \,c^{2}-a^{3}+4 a \alpha c \right ) \WhittakerM \left (\frac {2 \sqrt {a^{2}-4 \alpha c}\, n -a b +n a +2 \beta c -a}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )}{2 \left (a^{2}-4 \alpha c \right )^{\frac {3}{2}} x c \left (\WhittakerW \left (-\frac {a b -n a -2 \beta c +a}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1}+\WhittakerM \left (-\frac {a b -n a -2 \beta c +a}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right )} \]

✓ Solution by Mathematica

Time used: 3.691 (sec). Leaf size: 1837

DSolve[x^2*y'[x]==c*x^2*y[x]^2+(a*x^n+b)*x*y[x]+\[Alpha]*x^(2*n)+\[Beta]*x^n+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-\left (\left (-\left (\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )\right ) c_1 \text {HypergeometricU}\left (\frac {\left (3 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (6 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {2 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right ) x^n\right )-n \left (-a^2 n x^n+4 c n \alpha x^n+a n \sqrt {a^2-4 c \alpha } x^n+\sqrt {a^2-4 c \alpha } \left (b n+n+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )\right ) c_1 \text {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )-n \left (\left (-a^2 n x^n+4 c n \alpha x^n+a n \sqrt {a^2-4 c \alpha } x^n+\sqrt {a^2-4 c \alpha } \left (b n+n+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )\right ) L_{\frac {-\left (\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )}}^{\frac {\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )-2 n x^n \left (a^2-4 c \alpha \right ) L_{\frac {-\left (\left (3 n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (6 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )}}^{\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )\right )}{2 c n^2 x \sqrt {a^2-4 c \alpha } \left (c_1 \text {HypergeometricU}\left (\frac {\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2+(b-n+1) n \sqrt {a^2-4 c \alpha } a-2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )},\frac {n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2},\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )+L_{\frac {-\left (\left (n^2+\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right ) a^2\right )+n (-b+n-1) \sqrt {a^2-4 c \alpha } a+2 c \left (2 \alpha n^2+\sqrt {a^2-4 c \alpha } \beta n+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}\right )}{2 n^2 \left (a^2-4 c \alpha \right )}}^{\frac {\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}}{n^2}}\left (\frac {x^n \sqrt {a^2-4 c \alpha }}{n}\right )\right )} \\ y(x)\to \frac {-\frac {x^n \left (2 c \left (\beta n \sqrt {a^2-4 \alpha c}+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+2 \alpha n^2\right )-\left (a^2 \left (\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+n^2\right )\right )+a n (-b+n-1) \sqrt {a^2-4 \alpha c}\right ) \text {HypergeometricU}\left (\frac {-2 c \left (\beta n \sqrt {a^2-4 \alpha c}+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+6 \alpha n^2\right )+a^2 \left (\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+3 n^2\right )+a n (b-n+1) \sqrt {a^2-4 \alpha c}}{2 n^2 \left (a^2-4 \alpha c\right )},\frac {\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+2 n^2}{n^2},\frac {x^n \sqrt {a^2-4 \alpha c}}{n}\right )}{\text {HypergeometricU}\left (\frac {-2 c \left (\beta n \sqrt {a^2-4 \alpha c}+2 \alpha \sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+2 \alpha n^2\right )+a^2 \left (\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+n^2\right )+a n (b-n+1) \sqrt {a^2-4 \alpha c}}{2 n^2 \left (a^2-4 \alpha c\right )},\frac {\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+n^2}{n^2},\frac {x^n \sqrt {a^2-4 \alpha c}}{n}\right )}-n \left (\sqrt {a^2-4 \alpha c} \left (\sqrt {n^2 \left (b^2+2 b-4 c \gamma +1\right )}+b n+n\right )+a n x^n \sqrt {a^2-4 \alpha c}-a^2 n x^n+4 \alpha c n x^n\right )}{2 c n^2 x \sqrt {a^2-4 \alpha c}} \\ \end{align*}