36.6.16 problem 16

Internal problem ID [6377]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 16
Date solved : Monday, January 27, 2025 at 01:58:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 20

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

\[ y = 1-t -\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{6} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

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

\[ y(t)\to \frac {t^5}{120}+\frac {t^4}{6}+\frac {t^3}{6}-\frac {t^2}{2}-t+1 \]