36.5.15 problem 16

Internal problem ID [6359]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 16
Date solved : Monday, January 27, 2025 at 01:58:07 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 57

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

\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 66

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

\[ y(x)\to c_1 \left (-\frac {x^5}{30}-\frac {x^4}{8}-\frac {x^3}{3}-\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^5}{24}+\frac {x^4}{6}+\frac {x^3}{2}+x^2+x\right ) \]