73.25.31 problem 35.5 (c)

Internal problem ID [15739]
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 : Tuesday, January 28, 2025 at 08:06:20 AM
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*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 52

Order:=6; 
dsolve(4*diff(y(x),x$2)+(4*x-3)/(x-1)^2*y(x)=0,y(x),type='series',x=1);
 

\[ y = \left (\left (c_{2} \ln \left (x -1\right )+c_{1} \right ) \left (1-\left (x -1\right )+\frac {1}{4} \left (x -1\right )^{2}-\frac {1}{36} \left (x -1\right )^{3}+\frac {1}{576} \left (x -1\right )^{4}-\frac {1}{14400} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+\left (2 \left (x -1\right )-\frac {3}{4} \left (x -1\right )^{2}+\frac {11}{108} \left (x -1\right )^{3}-\frac {25}{3456} \left (x -1\right )^{4}+\frac {137}{432000} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) c_{2} \right ) \sqrt {x -1} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 162

AsymptoticDSolveValue[4*D[y[x],{x,2}]+(4*x-3)/(x-1)^2*y[x]==0,y[x],{x,1,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {(x-1)^5}{14400}+\frac {1}{576} (x-1)^4-\frac {1}{36} (x-1)^3+\frac {1}{4} (x-1)^2-x+2\right ) \sqrt {x-1}+c_2 \left (\sqrt {x-1} \left (\frac {137 (x-1)^5}{432000}-\frac {25 (x-1)^4}{3456}+\frac {11}{108} (x-1)^3-\frac {3}{4} (x-1)^2+2 (x-1)\right )+\left (-\frac {(x-1)^5}{14400}+\frac {1}{576} (x-1)^4-\frac {1}{36} (x-1)^3+\frac {1}{4} (x-1)^2-x+2\right ) \sqrt {x-1} \log (x-1)\right ) \]