61.2.5 problem 5
Internal
problem
ID
[12011]
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
:
5
Date
solved
:
Tuesday, January 28, 2025 at 06:24:11 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 388
dsolve(diff(y(x),x)=y(x)^2+a*n*x^(n-1)-a^2*x^(2*n),y(x), singsol=all)
\[
y = \frac {-3 c_{1} \left (n +2\right ) \left (\left (\frac {1}{3} n^{2}+n +\frac {2}{3}\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}} \left (n +\frac {4}{3}\right )\right ) {\mathrm e}^{\frac {a \,x^{n} x}{n +1}} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+2 c_{1} \left (n +1\right ) {\mathrm e}^{\frac {a \,x^{n} x}{n +1}} \left (\left (-\frac {1}{2} n^{2}-\frac {3}{2} n -1\right ) x^{-\frac {3 n}{2}}+x a \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {n}{2}}+a x \,x^{\frac {n}{2}}\right )\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+2 c_{1} \left (n +2\right )^{2} \left (n +\frac {3}{2}\right ) x^{-\frac {3 n}{2}} {\mathrm e}^{\frac {2 a \,x^{n} x}{n +1}} \left (-\frac {2 a \,x^{n} x}{n +1}\right )^{\frac {3 n +4}{2 n +2}}+2 x^{2} a \,x^{n}}{2 x \left (-\frac {{\mathrm e}^{\frac {a \,x^{n} x}{n +1}} x^{-\frac {3 n}{2}} c_{1} \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )}{2}+c_{1} \left (n +1\right ) {\mathrm e}^{\frac {a \,x^{n} x}{n +1}} \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}}\right ) \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+x \right )}
\]
✓ Solution by Mathematica
Time used: 1.005 (sec). Leaf size: 227
DSolve[D[y[x],x]==y[x]^2+a*n*x^(n-1)-a^2*x^(2*n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}} \left (a x^n-c_1 e^{\frac {2 a x^{n+1}}{n+1}}\right )-a c_1 x^{n+1} \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}{2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}-c_1 x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )} \\
y(x)\to \frac {2^{\frac {1}{n+1}} (n+1) e^{\frac {2 a x^{n+1}}{n+1}} \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}}{x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}+a x^n \\
\end{align*}