36.6.1 problem 1

Internal problem ID [6362]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 1
Date solved : Monday, January 27, 2025 at 01:58:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1+x \right ) y^{\prime \prime }-3 x y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 76

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

\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}-\frac {5 \left (x -1\right )^{4}}{48}-\frac {7 \left (x -1\right )^{5}}{240}\right ) y \left (1\right )+\left (x -1+\frac {3 \left (x -1\right )^{2}}{4}+\frac {\left (x -1\right )^{3}}{3}+\frac {\left (x -1\right )^{4}}{6}+\frac {7 \left (x -1\right )^{5}}{120}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (-\frac {7}{240} (x-1)^5-\frac {5}{48} (x-1)^4-\frac {1}{6} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (\frac {7}{120} (x-1)^5+\frac {1}{6} (x-1)^4+\frac {1}{3} (x-1)^3+\frac {3}{4} (x-1)^2+x-1\right ) \]