61.29.38 problem 147
Internal
problem
ID
[12647]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-4
Problem
number
:
147
Date
solved
:
Tuesday, January 28, 2025 at 08:03:09 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+x \left (2 a \,x^{n}+b \right ) y^{\prime }+\left (a^{2} x^{2 n}+a \left (b +n -1\right ) x^{n}+\alpha \,x^{2 m}+\beta \,x^{m}+\gamma \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.608 (sec). Leaf size: 111
dsolve(x^2*diff(y(x),x$2)+x*(2*a*x^n+b)*diff(y(x),x)+(a^2*x^(2*n)+a*(b+n-1)*x^n+alpha*x^(2*m)+beta*x^m+gamma)*y(x)=0,y(x), singsol=all)
\[
y = x^{-\frac {b}{2}} x^{-\frac {m}{2}} \sqrt {x}\, {\mathrm e}^{-\frac {a \,x^{n}}{n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i \beta }{2 m \sqrt {\alpha }}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 m}, \frac {2 i \sqrt {\alpha }\, x^{m}}{m}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i \beta }{2 m \sqrt {\alpha }}, \frac {\sqrt {b^{2}-2 b -4 \gamma +1}}{2 m}, \frac {2 i \sqrt {\alpha }\, x^{m}}{m}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.791 (sec). Leaf size: 291
DSolve[x^2*D[y[x],{x,2}]+x*(2*a*x^n+b)*D[y[x],x]+(a^2*x^(2*n)+a*(b+n-1)*x^n+\[Alpha]*x^(2*m)+\[Beta]*x^m+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to x^{\frac {1}{2}-\frac {m}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}+1\right )} \left (x^n\right )^{-\frac {b}{2 n}} \left (x^m\right )^{\frac {1}{2} \left (\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}+1\right )} e^{-\frac {a x^n}{n}+\frac {i \sqrt {\alpha } x^m}{m}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {m^2-\frac {i \beta m}{\sqrt {\alpha }}+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 m^2},\frac {m^2+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2},-\frac {2 i x^m \sqrt {\alpha }}{m}\right )+c_2 L_{-\frac {m^2-\frac {i \beta m}{\sqrt {\alpha }}+\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{2 m^2}}^{\frac {\sqrt {m^2 \left (b^2-2 b-4 \gamma +1\right )}}{m^2}}\left (-\frac {2 i x^m \sqrt {\alpha }}{m}\right )\right )
\]