14.12.1 problem 1

Internal problem ID [2611]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8. Series solutions. Excercises page 197
Problem number : 1
Date solved : Monday, January 27, 2025 at 06:04:34 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 39

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

\[ y = \left (1-\frac {1}{2} t^{2}+\frac {1}{8} t^{4}\right ) y \left (0\right )+\left (t -\frac {1}{3} t^{3}+\frac {1}{15} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 42

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

\[ y(t)\to c_2 \left (\frac {t^5}{15}-\frac {t^3}{3}+t\right )+c_1 \left (\frac {t^4}{8}-\frac {t^2}{2}+1\right ) \]