83.30.11 problem 11
Internal
problem
ID
[19319]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
VII.
Exact
differential
equations
and
certain
particular
forms
of
equations.
Exercise
VII
(D)
at
page
109
Problem
number
:
11
Date
solved
:
Monday, March 31, 2025 at 07:06:46 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} y^{\prime }-x y^{\prime \prime }-\frac {a^{2} y^{\prime }}{x}+\frac {x^{2}}{a}&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 52
ode:=diff(y(x),x)-x*diff(diff(y(x),x),x)-a^2/x*diff(y(x),x)+x^2/a = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\int x \left (\operatorname {Ei}_{1}\left (\frac {a^{2}}{x}\right ) {\mathrm e}^{\frac {a^{2}}{x}} a^{2}-{\mathrm e}^{\frac {a^{2}}{x}} c_1 a -x \right )d x}{a}+c_2
\]
✓ Mathematica. Time used: 0.166 (sec). Leaf size: 127
ode=D[y[x],x]-x*D[y[x],{x,2}]-a^2/x*D[y[x],x]+x^2/a==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {-6 a^2 x^2 G_{2,3}^{3,1}\left (\frac {a^2}{x}| \begin {array}{c} 0,3 \\ 0,0,2 \\ \end {array} \right )+3 a c_1 x e^{\frac {a^2}{x}} \left (a^2+x\right )+3 a^6 \left (\log \left (-\frac {a^2}{x}\right )-2 \log \left (\frac {a^2}{x}\right )-\log \left (-\frac {x}{a^2}\right )\right ) \Gamma \left (-2,-\frac {a^2}{x}\right )-3 a^5 c_1 \operatorname {ExpIntegralEi}\left (\frac {a^2}{x}\right )+2 x^3}{6 a}+c_2
\]
✓ Sympy. Time used: 57.548 (sec). Leaf size: 258
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a**2*Derivative(y(x), x)/x - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) + x**2/a,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + \begin {cases} \frac {C_{1} a \left (- \frac {a^{4} \operatorname {Ei}{\left (\frac {a^{2}}{x} \right )}}{2} + \frac {a^{2} x e^{\frac {a^{2}}{x}}}{2} + \frac {x^{2} e^{\frac {a^{2}}{x}}}{2}\right ) + a^{2} \int x e^{\frac {a^{2}}{x}} \operatorname {Ei}{\left (\frac {a^{2} e^{i \pi }}{x} \right )}\, dx + \frac {x^{3}}{3}}{a} & \text {for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac {2 \left (C_{1} a \left (\frac {a^{4} \log {\left (- a^{2} + 2 x \right )}}{8} + \frac {a^{2} x}{4} + \frac {x^{2}}{4}\right ) + a^{2} \int \frac {x^{2} \operatorname {Ei}{\left (\frac {a^{2} e^{i \pi }}{x} \right )}}{- a^{2} + 2 x}\, dx + \int \frac {x^{3} e^{- \frac {a^{2}}{x}}}{- a^{2} + 2 x}\, dx\right )}{a} & \text {for}\: a = 0 \\- 2 a \left (C_{1} a \int \frac {x^{2} e^{\frac {a^{2}}{x}}}{a^{4} e^{\frac {a^{2}}{x}} + 2 x^{2}}\, dx + a^{2} \int \frac {x^{2} e^{\frac {a^{2}}{x}} \operatorname {Ei}{\left (\frac {a^{2} e^{i \pi }}{x} \right )}}{a^{4} e^{\frac {a^{2}}{x}} + 2 x^{2}}\, dx + \int \frac {x^{3}}{a^{4} e^{\frac {a^{2}}{x}} + 2 x^{2}}\, dx\right ) & \text {otherwise} \end {cases}
\]