36.5.5 problem 5

Internal problem ID [6349]
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 : 5
Date solved : Monday, January 27, 2025 at 01:57:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (t^{2}-t -2\right ) x^{\prime \prime }+\left (1+t \right ) x^{\prime }-\left (t -2\right ) x&=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((t^2-t-2)*diff(x(t),t$2)+(t+1)*diff(x(t),t)-(t-2)*x(t)=0,x(t),type='series',t=0);
 

\[ x \left (t \right ) = \left (1+\frac {1}{2} t^{2}-\frac {1}{12} t^{3}+\frac {13}{96} t^{4}-\frac {1}{16} t^{5}\right ) x \left (0\right )+\left (t +\frac {1}{4} t^{2}+\frac {1}{4} t^{3}-\frac {1}{96} t^{4}+\frac {31}{480} t^{5}\right ) D\left (x \right )\left (0\right )+O\left (t^{6}\right ) \]

✓ Solution by Mathematica

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

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

\[ x(t)\to c_1 \left (-\frac {t^5}{16}+\frac {13 t^4}{96}-\frac {t^3}{12}+\frac {t^2}{2}+1\right )+c_2 \left (\frac {31 t^5}{480}-\frac {t^4}{96}+\frac {t^3}{4}+\frac {t^2}{4}+t\right ) \]