14.15.2 problem 2

Internal problem ID [2667]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.3, Equal roots and roots differing by an integer. Excercises page 223
Problem number : 2
Date solved : Monday, January 27, 2025 at 06:05:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }-t \left (1+t \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 46

Order:=6; 
dsolve(t^2*diff(y(t),t$2)-t*(1+t)*diff(y(t),t)+y(t)=0,y(t),type='series',t=0);
 

\[ y = t \left (\left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (-t -\frac {3}{4} t^{2}-\frac {11}{36} t^{3}-\frac {25}{288} t^{4}-\frac {137}{7200} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 112

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

\[ y(t)\to c_1 t \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right )+c_2 \left (t \left (-\frac {137 t^5}{7200}-\frac {25 t^4}{288}-\frac {11 t^3}{36}-\frac {3 t^2}{4}-t\right )+t \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right ) \log (t)\right ) \]