55.29.48 problem 108
Internal
problem
ID
[13881]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-3
Problem
number
:
108
Date
solved
:
Friday, October 03, 2025 at 06:55:22 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.044 (sec). Leaf size: 248
ode:=(a__2*x+b__2)*diff(diff(y(x),x),x)+(a__1*x+b__1)*diff(y(x),x)+(a__0*x+b__0)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\left (a_{1} +\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\right ) x}{2 a_{2}}} \left (a_{2} x +b_{2} \right )^{\frac {a_{1} b_{2} +a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}} \left (\operatorname {KummerU}\left (\frac {\left (a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1} \right ) \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}-2 a_{2}^{2} b_{0} +\left (2 a_{0} b_{2} +a_{1} b_{1} \right ) a_{2} -a_{1}^{2} b_{2}}{2 \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2}^{2}}, \frac {a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) c_2 +\operatorname {KummerM}\left (\frac {\left (a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1} \right ) \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}-2 a_{2}^{2} b_{0} +\left (2 a_{0} b_{2} +a_{1} b_{1} \right ) a_{2} -a_{1}^{2} b_{2}}{2 \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2}^{2}}, \frac {a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 0.167 (sec). Leaf size: 301
ode=(a2*x+b2)*D[y[x],{x,2}]+(a1*x+b1)*D[y[x],x]+(a0*x+b0)*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} (\text {a2} x+\text {b2})^{\frac {\text {a1} \text {b2}+\text {a2}^2-\text {a2} \text {b1}}{\text {a2}^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {2 \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {b0}\right ) \text {a2}^2+\left (\text {a1} \text {b1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \text {b1}+2 \text {a0} \text {b2}\right ) \text {a2}+\text {a1} \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {a1}\right ) \text {b2}}{2 \text {a2}^2 \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}},-\frac {\text {b1}}{\text {a2}}+\frac {\text {a1} \text {b2}}{\text {a2}^2}+2,\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} (\text {b2}+\text {a2} x)}{\text {a2}^2}\right )+c_2 L_{\frac {-2 \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}-\text {b0}\right ) \text {a2}^2+\left (-\text {a1} \text {b1}+\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} \text {b1}-2 \text {a0} \text {b2}\right ) \text {a2}+\text {a1} \left (\text {a1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}\right ) \text {b2}}{2 \text {a2}^2 \sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}}^{\frac {\text {a2}^2-\text {b1} \text {a2}+\text {a1} \text {b2}}{\text {a2}^2}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} (\text {b2}+\text {a2} x)}{\text {a2}^2}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a__0 = symbols("a__0")
a__1 = symbols("a__1")
a__2 = symbols("a__2")
b__0 = symbols("b__0")
b__1 = symbols("b__1")
b__2 = symbols("b__2")
y = Function("y")
ode = Eq((a__0*x + b__0)*y(x) + (a__1*x + b__1)*Derivative(y(x), x) + (a__2*x + b__2)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False