61.27.37 problem 47
Internal
problem
ID
[12547]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-2
Problem
number
:
47
Date
solved
:
Tuesday, January 28, 2025 at 08:02:15 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 1.573 (sec). Leaf size: 171
dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)+(b*x^(2*n)+c*x^(n-1))*y(x)=0,y(x), singsol=all)
\[
y = {\mathrm e}^{-\frac {x^{n +1} \left (a +\sqrt {a^{2}-4 b}\right )}{2 n +2}} x \left (\operatorname {KummerM}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right ) c_{1} +\operatorname {KummerU}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right ) c_{2} \right )
\]
✓ Solution by Mathematica
Time used: 0.310 (sec). Leaf size: 333
DSolve[D[y[x],{x,2}]+a*x^n*D[y[x],x]+(b*x^(2*n)+c*x^(n-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} \exp \left (-\frac {1}{2} x^{n+1} \left (\frac {\sqrt {a^2-4 b}}{\sqrt {(n+1)^2}}+\frac {a}{n+1}\right )\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {n \left (\sqrt {(n+1)^2} a^2+\sqrt {a^2-4 b} (n+1) a-4 b \sqrt {(n+1)^2}\right )-2 \sqrt {a^2-4 b} c (n+1)}{2 \left (a^2-4 b\right ) (n+1) \sqrt {(n+1)^2}},\frac {n}{n+1},\frac {\sqrt {a^2-4 b} x^{n+1}}{\sqrt {(n+1)^2}}\right )+c_2 L_{\frac {2 \sqrt {a^2-4 b} c (n+1)-n \left (\sqrt {(n+1)^2} a^2+\sqrt {a^2-4 b} (n+1) a-4 b \sqrt {(n+1)^2}\right )}{2 \left (a^2-4 b\right ) (n+1) \sqrt {(n+1)^2}}}^{-\frac {1}{n+1}}\left (\frac {\sqrt {a^2-4 b} x^{n+1}}{\sqrt {(n+1)^2}}\right )\right )
\]