52.2.33 problem 33(b)

Internal problem ID [8284]
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)
Date solved : Monday, January 27, 2025 at 03:47:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{t}+\lambda y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.022 (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);
 

\[ y = c_{1} \left (1-\frac {1}{6} \lambda \,t^{2}+\frac {1}{120} \lambda ^{2} t^{4}-\frac {1}{5040} \lambda ^{3} t^{6}+\operatorname {O}\left (t^{8}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} \lambda \,t^{2}+\frac {1}{24} \lambda ^{2} t^{4}-\frac {1}{720} \lambda ^{3} t^{6}+\operatorname {O}\left (t^{8}\right )\right )}{t} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 70

AsymptoticDSolveValue[D[y[t],{t,2}]+2/t*D[y[t],t]+\[Lambda]*y[t]==0,y[t],{t,0,"8"-1}]
 

\[ y(t)\to c_1 \left (-\frac {1}{720} \lambda ^3 t^5+\frac {\lambda ^2 t^3}{24}-\frac {\lambda t}{2}+\frac {1}{t}\right )+c_2 \left (-\frac {\lambda ^3 t^6}{5040}+\frac {\lambda ^2 t^4}{120}-\frac {\lambda t^2}{6}+1\right ) \]