23.3.476 problem 482
Internal
problem
ID
[6190]
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
:
482
Date
solved
:
Friday, October 03, 2025 at 01:56:47 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 \operatorname {a2} y+\left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (\operatorname {c0} \,x^{2}+\operatorname {b0} x +\operatorname {a0} \right ) y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 507
ode:=2*a2*y(x)+(b1*x+a1)*diff(y(x),x)+(c0*x^2+b0*x+a0)*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 3.033 (sec). Leaf size: 498
ode=2*a2*y[x] + (a1 + b1*x)*D[y[x],x] + (a0 + b0*x + c0*x^2)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_1 \operatorname {Hypergeometric2F1}\left (\frac {\text {b1}-\text {c0}+\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}}{2 \text {c0}},-\frac {-\text {b1}+\text {c0}+\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}}{2 \text {c0}},\frac {\text {b1} \left (\text {b0}+\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}\right )-2 \text {a1} \text {c0}}{2 \text {c0} \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}},\frac {\text {b0}+2 \text {c0} x+\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}{2 \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right )-c_2 2^{-\frac {\text {a1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\frac {\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {b1}}{2 \text {c0}}-1} \exp \left (-\frac {i \pi \left (\text {b1} \left (\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}+\text {b0}\right )-2 \text {a1} \text {c0}\right )}{2 \text {c0} \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right ) \left (\frac {\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}+\text {b0}+2 \text {c0} x}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\right )^{\frac {\text {a1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}-\frac {\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {b1}}{2 \text {c0}}+1} \operatorname {Hypergeometric2F1}\left (\frac {-\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {c0}-\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}+\frac {2 \text {a1} \text {c0}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}}{2 \text {c0}},\frac {-\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {c0}+\sqrt {(\text {b1}-\text {c0})^2-8 \text {a2} \text {c0}}+\frac {2 \text {a1} \text {c0}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}}{2 \text {c0}},-\frac {\frac {\text {b0} \text {b1}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}+\text {b1}-4 \text {c0}-\frac {2 \text {a1} \text {c0}}{\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}}{2 \text {c0}},\frac {\text {b0}+2 \text {c0} x+\sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}{2 \sqrt {\text {b0}^2-4 \text {a0} \text {c0}}}\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")
c0 = symbols("c0")
y = Function("y")
ode = Eq(2*a2*y(x) + (a1 + b1*x)*Derivative(y(x), x) + (a0 + b0*x + c0*x**2)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False