23.3.242 problem 244
Internal
problem
ID
[5956]
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
:
244
Date
solved
:
Friday, October 03, 2025 at 01:45:36 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 (\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 248
ode:=(b2*x+a2)*y(x)+(b1*x+a1)*diff(y(x),x)+(b0*x+a0)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\left (\operatorname {b1} +\sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}\right ) x}{2 \operatorname {b0}}} \left (\operatorname {b0} x +\operatorname {a0} \right )^{\frac {\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} +\operatorname {b0}^{2}}{\operatorname {b0}^{2}}} \left (\operatorname {KummerU}\left (\frac {\left (\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} +2 \operatorname {b0}^{2}\right ) \sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}-2 \operatorname {a2} \,\operatorname {b0}^{2}+\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {b0} -\operatorname {a0} \,\operatorname {b1}^{2}}{2 \sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}\, \operatorname {b0}^{2}}, \frac {\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} +2 \operatorname {b0}^{2}}{\operatorname {b0}^{2}}, \frac {\sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}\, \left (\operatorname {b0} x +\operatorname {a0} \right )}{\operatorname {b0}^{2}}\right ) c_2 +\operatorname {KummerM}\left (\frac {\left (\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} +2 \operatorname {b0}^{2}\right ) \sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}-2 \operatorname {a2} \,\operatorname {b0}^{2}+\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {b0} -\operatorname {a0} \,\operatorname {b1}^{2}}{2 \sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}\, \operatorname {b0}^{2}}, \frac {\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} +2 \operatorname {b0}^{2}}{\operatorname {b0}^{2}}, \frac {\sqrt {-4 \operatorname {b2} \operatorname {b0} +\operatorname {b1}^{2}}\, \left (\operatorname {b0} x +\operatorname {a0} \right )}{\operatorname {b0}^{2}}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 0.137 (sec). Leaf size: 307
ode=(a2 + b2*x)*y[x] + (a1 + b1*x)*D[y[x],x] + (a0 + b0*x)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\frac {x \left (\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}+\text {b1}\right )}{2 \text {b0}}} (\text {a0}+\text {b0} x)^{\frac {\text {a0} \text {b1}-\text {a1} \text {b0}+\text {b0}^2}{\text {b0}^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-2 \text {a2} \text {b0}^2+2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}} \text {b0}^2+2 \text {a0} \text {b2} \text {b0}+\text {a1} \left (\text {b1}-\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}\right ) \text {b0}-\text {a0} \text {b1}^2+\text {a0} \text {b1} \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}},-\frac {\text {a1}}{\text {b0}}+\frac {\text {a0} \text {b1}}{\text {b0}^2}+2,\frac {\sqrt {\text {b1}^2-4 \text {b0} \text {b2}} (\text {a0}+\text {b0} x)}{\text {b0}^2}\right )+c_2 L_{\frac {2 \text {a2} \text {b0}^2-2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}} \text {b0}^2-2 \text {a0} \text {b2} \text {b0}+\text {a1} \left (\sqrt {\text {b1}^2-4 \text {b0} \text {b2}}-\text {b1}\right ) \text {b0}+\text {a0} \text {b1}^2-\text {a0} \text {b1} \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}{2 \text {b0}^2 \sqrt {\text {b1}^2-4 \text {b0} \text {b2}}}}^{\frac {\text {b0}^2-\text {a1} \text {b0}+\text {a0} \text {b1}}{\text {b0}^2}}\left (\frac {\sqrt {\text {b1}^2-4 \text {b0} \text {b2}} (\text {a0}+\text {b0} x)}{\text {b0}^2}\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a0 = symbols("a0")
a1 = symbols("a1")
a2 = symbols("a2")
b0 = symbols("b0")
b1 = symbols("b1")
b2 = symbols("b2")
y = Function("y")
ode = Eq((a0 + b0*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