67.25.31 problem 35.5 (c)

Internal problem ID [17026]
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 : Thursday, October 02, 2025 at 01:42:03 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: 0.005 (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: 0.390 (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)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x - 1} \left (- x + \frac {\left (x - 1\right )^{4}}{576} - \frac {\left (x - 1\right )^{3}}{36} + \frac {\left (x - 1\right )^{2}}{4} + 2\right ) + O\left (x^{6}\right ) \]