Internal problem ID [13683]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {4 y^{\prime \prime }+\frac {\left (4 x -3\right ) y}{\left (x -1\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 77
Order:=6; dsolve(4*diff(y(x),x$2)+(4*x-3)/(x-1)^2*y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \sqrt {-1+x}\, \left (\left (\ln \left (-1+x \right ) c_{2} +c_{1} \right ) \left (1-\left (-1+x \right )+\frac {1}{4} \left (-1+x \right )^{2}-\frac {1}{36} \left (-1+x \right )^{3}+\frac {1}{576} \left (-1+x \right )^{4}-\frac {1}{14400} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )+\left (2 \left (-1+x \right )-\frac {3}{4} \left (-1+x \right )^{2}+\frac {11}{108} \left (-1+x \right )^{3}-\frac {25}{3456} \left (-1+x \right )^{4}+\frac {137}{432000} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 162
AsymptoticDSolveValue[4*y''[x]+(4*x-3)/(x-1)^2*y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (-\frac {(x-1)^5}{14400}+\frac {1}{576} (x-1)^4-\frac {1}{36} (x-1)^3+\frac {1}{4} (x-1)^2-x+2\right ) \sqrt {x-1}+c_2 \left (\sqrt {x-1} \left (\frac {137 (x-1)^5}{432000}-\frac {25 (x-1)^4}{3456}+\frac {11}{108} (x-1)^3-\frac {3}{4} (x-1)^2+2 (x-1)\right )+\left (-\frac {(x-1)^5}{14400}+\frac {1}{576} (x-1)^4-\frac {1}{36} (x-1)^3+\frac {1}{4} (x-1)^2-x+2\right ) \sqrt {x-1} \log (x-1)\right ) \]