Internal problem ID [4498]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (t^{2}-t -2\right ) x^{\prime \prime }+\left (t +1\right ) x^{\prime }-\left (t -2\right ) x=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 54
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);
\[ x \relax (t ) = \left (1+\frac {1}{2} t^{2}-\frac {1}{12} t^{3}+\frac {13}{96} t^{4}-\frac {1}{16} t^{5}\right ) x \relax (0)+\left (t +\frac {1}{4} t^{2}+\frac {1}{4} t^{3}-\frac {1}{96} t^{4}+\frac {31}{480} t^{5}\right ) D\relax (x )\relax (0)+O\left (t^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 70
AsymptoticDSolveValue[(t^2-t-2)*x''[t]+(t+1)*x'[t]-(t-2)*x[t]==0,x[t],{t,0,5}]
\[ x(t)\to c_1 \left (-\frac {t^5}{16}+\frac {13 t^4}{96}-\frac {t^3}{12}+\frac {t^2}{2}+1\right )+c_2 \left (\frac {31 t^5}{480}-\frac {t^4}{96}+\frac {t^3}{4}+\frac {t^2}{4}+t\right ) \]