Internal problem ID [5863]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 33(b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{t}+\lambda y=0} \end {gather*} With the expansion point for the power series method at \(t = 0\).
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 56
Order:=8; dsolve(diff(y(t),t$2)+2/t*diff(y(t),t)+lambda*y(t)=0,y(t),type='series',t=0);
\[ y \relax (t ) = c_{1} \left (1-\frac {1}{6} \lambda \,t^{2}+\frac {1}{120} \lambda ^{2} t^{4}-\frac {1}{5040} \lambda ^{3} t^{6}+\mathrm {O}\left (t^{8}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} \lambda \,t^{2}+\frac {1}{24} \lambda ^{2} t^{4}-\frac {1}{720} \lambda ^{3} t^{6}+\mathrm {O}\left (t^{8}\right )\right )}{t} \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 70
AsymptoticDSolveValue[y''[t]+2/t*y'[t]+\[Lambda]*y[t]==0,y[t],{t,0,7}]
\[ y(t)\to c_1 \left (-\frac {1}{720} \lambda ^3 t^5+\frac {\lambda ^2 t^3}{24}-\frac {\lambda t}{2}+\frac {1}{t}\right )+c_2 \left (-\frac {\lambda ^3 t^6}{5040}+\frac {\lambda ^2 t^4}{120}-\frac {\lambda t^2}{6}+1\right ) \]