61.27.27 problem 37
Internal
problem
ID
[12537]
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
:
37
Date
solved
:
Tuesday, January 28, 2025 at 08:02:12 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 1.032 (sec). Leaf size: 271
dsolve(diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+(alpha*x^2+beta*x+gamma)*y(x)=0,y(x), singsol=all)
\[
y = c_{1} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} \operatorname {csgn}\left (a \right ) x^{2}+3 a x b \,\operatorname {csgn}\left (a \right )+2 a^{2} x^{2}+3 a x b -12 \alpha \right )}{12 a}} \operatorname {HeunT}\left (\frac {3^{{2}/{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{{1}/{3}}}, -\frac {3 \left (a^{2}-a \beta +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{{1}/{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )+c_{2} {\mathrm e}^{-\frac {\operatorname {csgn}\left (a \right ) x \left (2 a^{2} \operatorname {csgn}\left (a \right ) x^{2}+3 a x b \,\operatorname {csgn}\left (a \right )-2 a^{2} x^{2}-3 a x b +12 \alpha \right )}{12 a}} \operatorname {HeunT}\left (\frac {3^{{2}/{3}} \left (2 a^{2} \gamma -a b \beta +\alpha \,b^{2}+2 \alpha ^{2}\right )}{2 a^{2} \left (a^{2}\right )^{{1}/{3}}}, \frac {3 \left (a^{2}-a \beta +b \alpha \right ) \operatorname {csgn}\left (a \right )}{a^{2}}, -\frac {3^{{1}/{3}} \left (b^{2}+8 \alpha \right )}{4 \left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )
\]
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[D[y[x],{x,2}]+(a*x^2+b*x)*D[y[x],x]+(\[Alpha]*x^2+\[Beta]*x+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved