54.3.131 problem 1145
Internal
problem
ID
[12426]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1145
Date
solved
:
Friday, October 03, 2025 at 03:19:09 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.044 (sec). Leaf size: 248
ode:=(a2*x+b2)*diff(diff(y(x),x),x)+(a1*x+b1)*diff(y(x),x)+(a0*x+b0)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\left (\operatorname {a1} +\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\right ) x}{2 \operatorname {a2}}} \left (\operatorname {a2} x +\operatorname {b2} \right )^{\frac {\operatorname {a1} \operatorname {b2} +\operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}} \left (\operatorname {KummerU}\left (\frac {\left (\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1} \right ) \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}-2 \operatorname {a2}^{2} \operatorname {b0} +\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {a2} -\operatorname {a1}^{2} \operatorname {b2}}{2 \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2}^{2}}, \frac {\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) c_2 +\operatorname {KummerM}\left (\frac {\left (\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1} \right ) \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}-2 \operatorname {a2}^{2} \operatorname {b0} +\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {a2} -\operatorname {a1}^{2} \operatorname {b2}}{2 \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2}^{2}}, \frac {\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) c_1 \right )
\]
✓ Mathematica. Time used: 1.156 (sec). Leaf size: 328
ode=(b0 + a0*x)*y[x] + (b1 + a1*x)*D[y[x],x] + (b2 + a2*x)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \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 ) \exp \left (\int _1^x\frac {2 \text {a2}^2-\left (2 \text {b1}+\left (\text {a1}+\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}\right ) K[1]\right ) \text {a2}+\left (\text {a1}-\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}\right ) \text {b2}}{2 \text {a2} (\text {b2}+\text {a2} K[1])}dK[1]\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*x + b0)*y(x) + (a1*x + b1)*Derivative(y(x), x) + (a2*x + b2)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False