61.2.26 problem 26
Internal
problem
ID
[12032]
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
:
26
Date
solved
:
Monday, January 27, 2025 at 11:51:50 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a \,x^{n} y+b \,x^{n -1} \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 267
dsolve(diff(y(x),x)=y(x)^2+a*x^n*y(x)+b*x^(n-1),y(x), singsol=all)
\[
y = \frac {-a \left (n +1\right ) \left (a -b \right ) \operatorname {KummerM}\left (\frac {a \left (n +2\right )-b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )+\left (\left (a -b \right ) c_{1} \operatorname {KummerU}\left (\frac {2+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )-a \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )\right ) \left (n +1\right )\right ) b}{a^{2} \left (n +1\right ) x \left (\operatorname {KummerU}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right ) c_{1} +\operatorname {KummerM}\left (\frac {a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.513 (sec). Leaf size: 453
DSolve[D[y[x],x]==y[x]^2+a*x^n*y[x]+b*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\left (x^n\right )^{\frac {1}{n}} \left (-(-1)^{\frac {1}{n+1}} n (n+2) a^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+x^n \left (-(-1)^{\frac {1}{n+1}} n (a-b) a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+2 a-b}{n a+a},\frac {2 n+3}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+b c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} (n+2) \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )\right )}{n (n+2) x \left ((-1)^{\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a-b}{n a+a},\frac {n+2}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \left (\frac {1}{n}+1\right )^{\frac {1}{n+1}} n^{\frac {1}{n+1}} \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )\right )} \\
y(x)\to \frac {b x^{n-1} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {n a+a-b}{n a+a},\frac {2 n+1}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )}{n \operatorname {Hypergeometric1F1}\left (-\frac {b}{n a+a},\frac {n}{n+1},\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )} \\
\end{align*}