13.12.15 problem 14

Internal problem ID [2427]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 14
Date solved : Monday, January 27, 2025 at 05:52:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y t&=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 )&=2 \end{align*}

✓ Solution by Maple

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

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

\[ y = -1+2 t -t^{2}+\frac {1}{2} t^{3}-\frac {7}{24} t^{4}+\frac {13}{120} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

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

\[ y(t)\to \frac {13 t^5}{120}-\frac {7 t^4}{24}+\frac {t^3}{2}-t^2+2 t-1 \]