46.2.33 problem 33(b)

Internal problem ID [9570]
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 : Tuesday, September 30, 2025 at 06:20:54 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*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 56

Order:=8; 
ode:=diff(diff(y(t),t),t)+2/t*diff(y(t),t)+lambda*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = c_1 \left (1-\frac {1}{6} \lambda \,t^{2}+\frac {1}{120} \lambda ^{2} t^{4}-\frac {1}{5040} \lambda ^{3} t^{6}+\operatorname {O}\left (t^{8}\right )\right )+\frac {c_2 \left (1-\frac {1}{2} \lambda \,t^{2}+\frac {1}{24} \lambda ^{2} t^{4}-\frac {1}{720} \lambda ^{3} t^{6}+\operatorname {O}\left (t^{8}\right )\right )}{t} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 70

ode=D[y[t],{t,2}]+2/t*D[y[t],t]+\[Lambda]*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,7}]
 

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

✓ Sympy. Time used: 0.326 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_*y(t) + Derivative(y(t), (t, 2)) + 2*Derivative(y(t), t)/t,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (t \right )} = C_{2} \left (- \frac {\lambda _{}^{3} t^{6}}{5040} + \frac {\lambda _{}^{2} t^{4}}{120} - \frac {\lambda _{} t^{2}}{6} + 1\right ) + \frac {C_{1} \left (\frac {\lambda _{}^{4} t^{8}}{40320} - \frac {\lambda _{}^{3} t^{6}}{720} + \frac {\lambda _{}^{2} t^{4}}{24} - \frac {\lambda _{} t^{2}}{2} + 1\right )}{t} + O\left (t^{8}\right ) \]