73.25.3 problem 35.2 (c)

Internal problem ID [15711]
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 : 35.2 (c)
Date solved : Tuesday, January 28, 2025 at 08:05:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -1\right )^{2} y^{\prime \prime }-5 \left (x -1\right ) y^{\prime }+9 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 59

Order:=6; 
dsolve((x-1)^2*diff(y(x),x$2)-5*(x-1)*diff(y(x),x)+9*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {9}{2} x^{2}+\frac {9}{2} x^{3}-\frac {3}{4} x^{4}-\frac {3}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {5}{2} x^{2}+\frac {11}{6} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 70

AsymptoticDSolveValue[(x-1)^2*D[y[x],{x,2}]-5*(x-1)*D[y[x],x]+9*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {3 x^5}{20}-\frac {3 x^4}{4}+\frac {9 x^3}{2}-\frac {9 x^2}{2}+1\right )+c_2 \left (-\frac {x^5}{20}-\frac {x^4}{4}+\frac {11 x^3}{6}-\frac {5 x^2}{2}+x\right ) \]