23.3.423 problem 428
Internal
problem
ID
[6137]
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
:
428
Date
solved
:
Friday, October 03, 2025 at 01:46:37 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 6 y-4 \left (a +x \right ) y^{\prime }+\left (\operatorname {a0} +x \right )^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 132
ode:=6*y(x)-4*(x+a)*diff(y(x),x)+(a0+x)^2*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = 32 \left (a -\operatorname {a0} \right ) \left (-\frac {\operatorname {a0}^{2}}{8}+\left (-\frac {a}{2}-\frac {3 x}{4}\right ) \operatorname {a0} +a^{2}+\frac {3 a x}{2}+\frac {3 x^{2}}{8}\right ) c_2 \,\operatorname {Ei}_{1}\left (\frac {4 a -4 \operatorname {a0}}{\operatorname {a0} +x}\right )-8 c_2 \left (\operatorname {a0} +x \right ) \left (-\frac {\operatorname {a0}^{2}}{8}+\left (-\frac {3 a}{4}-x \right ) \operatorname {a0} +a^{2}+\frac {5 a x}{4}+\frac {x^{2}}{8}\right ) {\mathrm e}^{\frac {-4 a +4 \operatorname {a0}}{\operatorname {a0} +x}}+\frac {8 \left (-\frac {\operatorname {a0}^{2}}{8}+\left (-\frac {a}{2}-\frac {3 x}{4}\right ) \operatorname {a0} +a^{2}+\frac {3 a x}{2}+\frac {3 x^{2}}{8}\right ) c_1}{3}
\]
✓ Mathematica. Time used: 0.501 (sec). Leaf size: 322
ode=6*y[x] - 4*(a + x)*D[y[x],x] + (a0 + x)^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{3} e^{-\frac {4 a}{\text {a0}+x}} \left (12 c_2 (a-\text {a0}) e^{\frac {4 a}{\text {a0}+x}} \left (8 a^2-4 a (\text {a0}-3 x)-\text {a0}^2-6 \text {a0} x+3 x^2\right ) \operatorname {ExpIntegralEi}\left (-\frac {4 (a-\text {a0})}{\text {a0}+x}\right )+8 a^2 \left (c_1 e^{\frac {4 a}{\text {a0}+x}}+3 c_2 e^{\frac {4 \text {a0}}{\text {a0}+x}} (\text {a0}+x)\right )-\text {a0}^2 c_1 e^{\frac {4 a}{\text {a0}+x}}-2 a \left (2 \text {a0} \left (c_1 e^{\frac {4 a}{\text {a0}+x}}-3 c_2 x e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )-3 x \left (2 c_1 e^{\frac {4 a}{\text {a0}+x}}+5 c_2 x e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )+9 \text {a0}^2 c_2 e^{\frac {4 \text {a0}}{\text {a0}+x}}\right )+3 c_1 x^2 e^{\frac {4 a}{\text {a0}+x}}-6 \text {a0} c_1 x e^{\frac {4 a}{\text {a0}+x}}-3 \text {a0}^3 c_2 e^{\frac {4 \text {a0}}{\text {a0}+x}}-27 \text {a0}^2 c_2 x e^{\frac {4 \text {a0}}{\text {a0}+x}}+3 c_2 x^3 e^{\frac {4 \text {a0}}{\text {a0}+x}}-21 \text {a0} c_2 x^2 e^{\frac {4 \text {a0}}{\text {a0}+x}}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
a0 = symbols("a0")
y = Function("y")
ode = Eq(-(4*a + 4*x)*Derivative(y(x), x) + (a0 + x)**2*Derivative(y(x), (x, 2)) + 6*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False