61.2.25 problem 25
Internal
problem
ID
[12031]
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
:
25
Date
solved
:
Monday, January 27, 2025 at 11:51:47 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \,x^{n} y+a \,x^{n -1} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 400
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+a*x^(n-1),y(x), singsol=all)
\[
y = \frac {-\left (-\frac {a \,x^{n} x}{n +1}\right )^{-\frac {n}{2 n +2}} {\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (n +1\right )^{2} \left (x^{n} a x -n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a \,x^{n} x}{n +1}\right )-2 n \left (-\frac {\left (n +1\right ) {\mathrm e}^{\frac {x^{n} a x}{2 n +2}} n \left (-\frac {a \,x^{n} x}{n +1}\right )^{-\frac {n}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a \,x^{n} x}{n +1}\right )}{2}+x \left (n +\frac {1}{2}\right ) a \,x^{n} \left (c_{1} x -{\mathrm e}^{\frac {a \,x^{n} x}{n +1}}\right )\right )}{x \left (\left (-\frac {a \,x^{n} x}{n +1}\right )^{-\frac {n}{2 n +2}} {\mathrm e}^{\frac {x^{n} a x}{2 n +2}} \left (n +1\right )^{2} \left (x^{n} a x -n \right ) \operatorname {WhittakerM}\left (\frac {-n -2}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a \,x^{n} x}{n +1}\right )+2 \left (-\frac {\left (n +1\right ) {\mathrm e}^{\frac {x^{n} a x}{2 n +2}} n \left (-\frac {a \,x^{n} x}{n +1}\right )^{-\frac {n}{2 n +2}} \operatorname {WhittakerM}\left (\frac {n}{2 n +2}, \frac {2 n +1}{2 n +2}, -\frac {a \,x^{n} x}{n +1}\right )}{2}+a \,x^{2} c_{1} x^{n} \left (n +\frac {1}{2}\right )\right ) n \right )}
\]
✓ Solution by Mathematica
Time used: 0.786 (sec). Leaf size: 268
DSolve[D[y[x],x]==y[x]^2+a*x^n*y[x]+a*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {a x^{n+1}}{n+1}}}{(x K[2]+1)^2}-\int _1^x\left (\frac {a e^{\frac {a K[1]^{n+1}}{n+1}} K[1]^n}{K[1] K[2]+1}-\frac {a e^{\frac {a K[1]^{n+1}}{n+1}} K[2] K[1]^{n+1}}{(K[1] K[2]+1)^2}+\frac {2 e^{\frac {a K[1]^{n+1}}{n+1}} K[2]^2 K[1]}{(K[1] K[2]+1)^3}-\frac {2 e^{\frac {a K[1]^{n+1}}{n+1}} K[2]}{(K[1] K[2]+1)^2}\right )dK[1]\right )dK[2]+\int _1^x\left (-a e^{\frac {a K[1]^{n+1}}{n+1}} K[1]^{n-1}+\frac {a e^{\frac {a K[1]^{n+1}}{n+1}} y(x) K[1]^n}{K[1] y(x)+1}-\frac {e^{\frac {a K[1]^{n+1}}{n+1}} y(x)^2}{(K[1] y(x)+1)^2}\right )dK[1]=c_1,y(x)\right ]
\]