61.27.21 problem 31
Internal
problem
ID
[12531]
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
:
31
Date
solved
:
Tuesday, January 28, 2025 at 08:02:10 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (\alpha \,x^{2}+\beta x +\gamma \right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.142 (sec). Leaf size: 254
dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+(alpha*x^2+beta*x+gamma)*y(x)=0,y(x), singsol=all)
\[
y = {\mathrm e}^{-\frac {x \left (\left (a x +2 b \right ) \sqrt {a^{2}-4 \alpha }+x \left (a^{2}-4 \alpha \right )+2 a b -4 \beta \right )}{4 \sqrt {a^{2}-4 \alpha }}} \left (c_{2} \left (a^{2} x +a b -4 \alpha x -2 \beta \right ) \operatorname {hypergeom}\left (\left [\frac {3 \left (a^{2}-4 \alpha \right )^{{3}/{2}}+a^{3}-2 a^{2} \gamma +2 \left (b \beta -2 \alpha \right ) a +2 \left (-b^{2}+4 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {\left (a^{2} x +a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right )+\operatorname {hypergeom}\left (\left [\frac {\left (a^{2}-4 \alpha \right )^{{3}/{2}}+a^{3}-2 a^{2} \gamma +\left (2 b \beta -4 \alpha \right ) a +\left (-2 b^{2}+8 \gamma \right ) \alpha -2 \beta ^{2}}{4 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {\left (a^{2} x +a b -4 \alpha x -2 \beta \right )^{2}}{2 \left (a^{2}-4 \alpha \right )^{{3}/{2}}}\right ) c_{1} \right )
\]
✓ Solution by Mathematica
Time used: 0.240 (sec). Leaf size: 307
DSolve[D[y[x],{x,2}]+(a*x+b)*D[y[x],x]+(\[Alpha]*x^2+\[Beta]*x+\[Gamma])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \exp \left (-\frac {x \left (2 b \sqrt {a^2-4 \alpha }+a \left (x \sqrt {a^2-4 \alpha }+2 b\right )+a^2 x-4 (\beta +\alpha x)\right )}{4 \sqrt {a^2-4 \alpha }}\right ) \left (c_1 \operatorname {HermiteH}\left (\frac {-a^3-\left (\sqrt {a^2-4 \alpha }-2 \gamma \right ) a^2+(4 \alpha -2 b \beta ) a+2 \left (\alpha b^2+\beta ^2+2 \sqrt {a^2-4 \alpha } \alpha -4 \alpha \gamma \right )}{2 \left (a^2-4 \alpha \right )^{3/2}},\frac {x a^2+b a-2 (2 x \alpha +\beta )}{\sqrt {2} \left (a^2-4 \alpha \right )^{3/4}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3+\left (\sqrt {a^2-4 \alpha }-2 \gamma \right ) a^2+(2 b \beta -4 \alpha ) a-2 \left (\alpha b^2+\beta ^2+2 \sqrt {a^2-4 \alpha } \alpha -4 \alpha \gamma \right )}{4 \left (a^2-4 \alpha \right )^{3/2}},\frac {1}{2},\frac {\left (x a^2+b a-2 (2 x \alpha +\beta )\right )^2}{2 \left (a^2-4 \alpha \right )^{3/2}}\right )\right )
\]