73.25.31 problem 35.5 (c)
Internal
problem
ID
[15668]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
35.
Modified
Power
series
solutions
and
basic
method
of
Frobenius.
Additional
Exercises.
page
715
Problem
number
:
35.5
(c)
Date
solved
:
Monday, March 31, 2025 at 01:44:32 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 y^{\prime \prime }+\frac {\left (4 x -3\right ) y}{\left (x -1\right )^{2}}&=0 \end{align*}
Using series method with expansion around
\begin{align*} 1 \end{align*}
✓ Maple. Time used: 0.016 (sec). Leaf size: 52
Order:=6;
ode:=4*diff(diff(y(x),x),x)+(4*x-3)/(x-1)^2*y(x) = 0;
dsolve(ode,y(x),type='series',x=1);
\[
y = \sqrt {x -1}\, \left (\left (c_2 \ln \left (x -1\right )+c_1 \right ) \left (1-\left (x -1\right )+\frac {1}{4} \left (x -1\right )^{2}-\frac {1}{36} \left (x -1\right )^{3}+\frac {1}{576} \left (x -1\right )^{4}-\frac {1}{14400} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+\left (2 \left (x -1\right )-\frac {3}{4} \left (x -1\right )^{2}+\frac {11}{108} \left (x -1\right )^{3}-\frac {25}{3456} \left (x -1\right )^{4}+\frac {137}{432000} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) c_2 \right )
\]
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 162
ode=4*D[y[x],{x,2}]+(4*x-3)/(x-1)^2*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
\[
y(x)\to c_1 \left (-\frac {(x-1)^5}{14400}+\frac {1}{576} (x-1)^4-\frac {1}{36} (x-1)^3+\frac {1}{4} (x-1)^2-x+2\right ) \sqrt {x-1}+c_2 \left (\sqrt {x-1} \left (\frac {137 (x-1)^5}{432000}-\frac {25 (x-1)^4}{3456}+\frac {11}{108} (x-1)^3-\frac {3}{4} (x-1)^2+2 (x-1)\right )+\left (-\frac {(x-1)^5}{14400}+\frac {1}{576} (x-1)^4-\frac {1}{36} (x-1)^3+\frac {1}{4} (x-1)^2-x+2\right ) \sqrt {x-1} \log (x-1)\right )
\]
✓ Sympy. Time used: 1.034 (sec). Leaf size: 37
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*Derivative(y(x), (x, 2)) + (4*x - 3)*y(x)/(x - 1)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=1,n=6)
\[
y{\left (x \right )} = C_{1} \sqrt {x - 1} \left (- x + \frac {\left (x - 1\right )^{4}}{576} - \frac {\left (x - 1\right )^{3}}{36} + \frac {\left (x - 1\right )^{2}}{4} + 2\right ) + O\left (x^{6}\right )
\]