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 (x^{n} a +b \right ) x y-\alpha \,x^{2 n}-\beta \,x^{n}-\gamma =0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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 {b a -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 -b a +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 \alpha c n a +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 {b a -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 {a^{2}-4 \alpha c}\, \sqrt {b^{2}-4 c \gamma +2 b +1}\, a^{2}-4 \sqrt {a^{2}-4 \alpha c}\, \sqrt {b^{2}-4 c \gamma +2 b +1}\, \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 \alpha c n a -8 \alpha \beta \,c^{2}-a^{3}+4 a \alpha c \right ) \WhittakerM \left (\frac {2 \sqrt {a^{2}-4 \alpha c}\, n -b a +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 {b a -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 {b a -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: 2.004 (sec). Leaf size: 2380
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]
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