61.2.53 problem 53
Internal
problem
ID
[12059]
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
Date
solved
:
Monday, January 27, 2025 at 11:55:15 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{2} y^{\prime }&=c \,x^{2} y^{2}+\left (a \,x^{n}+b \right ) x y+\alpha \,x^{2 n}+\beta \,x^{n}+\gamma \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 560
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)
\[
y = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}+\sqrt {a^{2}-4 \alpha c}\, n +\left (n -b -1\right ) a +2 \beta c \right ) \operatorname {WhittakerM}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )-2 \operatorname {WhittakerW}\left (-\frac {-2 \sqrt {a^{2}-4 \alpha c}\, n +a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} n \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) \left (\left (a \,x^{n}+b -n +1\right ) \sqrt {a^{2}-4 \alpha c}+\left (a^{2}-4 \alpha c \right ) x^{n}+a \left (b -n +1\right )-2 \beta c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a \left (b -n +1\right )-2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}\, n}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2 n}, \frac {\sqrt {a^{2}-4 \alpha c}\, x^{n}}{n}\right )\right ) c x}
\]
✓ Solution by Mathematica
Time used: 2.182 (sec). Leaf size: 2380
DSolve[x^2*D[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],x,IncludeSingularSolutions -> True]
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