23.3.229 problem 231
Internal
problem
ID
[5943]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
3.
THE
DIFFERENTIAL
EQUATION
IS
LINEAR
AND
OF
SECOND
ORDER,
page
311
Problem
number
:
231
Date
solved
:
Friday, October 03, 2025 at 01:45:30 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (a +x \right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.032 (sec). Leaf size: 171
ode:=(b2*x+a2)*y(x)+(b1*x+a1)*diff(y(x),x)+(x+a)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (a +x \right )^{\operatorname {b1} a -\operatorname {a1} +1} {\mathrm e}^{-\frac {\left (\operatorname {b1} +\sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}\right ) x}{2}} \left (\operatorname {KummerU}\left (-\frac {\left (-\operatorname {b1} a +\operatorname {a1} -2\right ) \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}+a \,\operatorname {b1}^{2}-2 \operatorname {b2} a -\operatorname {a1} \operatorname {b1} +2 \operatorname {a2}}{2 \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}}, \operatorname {b1} a -\operatorname {a1} +2, \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}\, \left (a +x \right )\right ) c_2 +\operatorname {KummerM}\left (-\frac {\left (-\operatorname {b1} a +\operatorname {a1} -2\right ) \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}+a \,\operatorname {b1}^{2}-2 \operatorname {b2} a -\operatorname {a1} \operatorname {b1} +2 \operatorname {a2}}{2 \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}}, \operatorname {b1} a -\operatorname {a1} +2, \sqrt {\operatorname {b1}^{2}-4 \operatorname {b2}}\, \left (a +x \right )\right ) c_1 \right )
\]
✓ Mathematica. Time used: 0.107 (sec). Leaf size: 239
ode=(a2 + b2*x)*y[x] + (a1 + b1*x)*D[y[x],x] + (a + x)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\frac {1}{2} x \left (\sqrt {\text {b1}^2-4 \text {b2}}+\text {b1}\right )} (a+x)^{a \text {b1}-\text {a1}+1} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-a \text {b1}^2+\text {a1} \text {b1}+a \sqrt {\text {b1}^2-4 \text {b2}} \text {b1}-2 \text {a2}+2 a \text {b2}-\text {a1} \sqrt {\text {b1}^2-4 \text {b2}}+2 \sqrt {\text {b1}^2-4 \text {b2}}}{2 \sqrt {\text {b1}^2-4 \text {b2}}},-\text {a1}+a \text {b1}+2,\sqrt {\text {b1}^2-4 \text {b2}} (a+x)\right )+c_2 L_{\frac {a \text {b1}^2-\text {a1} \text {b1}-a \sqrt {\text {b1}^2-4 \text {b2}} \text {b1}+2 \text {a2}-2 a \text {b2}+\text {a1} \sqrt {\text {b1}^2-4 \text {b2}}-2 \sqrt {\text {b1}^2-4 \text {b2}}}{2 \sqrt {\text {b1}^2-4 \text {b2}}}}^{-\text {a1}+a \text {b1}+1}\left (\sqrt {\text {b1}^2-4 \text {b2}} (a+x)\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
a1 = symbols("a1")
a2 = symbols("a2")
b1 = symbols("b1")
b2 = symbols("b2")
y = Function("y")
ode = Eq((a + x)*Derivative(y(x), (x, 2)) + (a1 + b1*x)*Derivative(y(x), x) + (a2 + b2*x)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False