14.12.12 problem 12 (b)

Internal problem ID [2622]
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 : 12 (b)
Date solved : Monday, January 27, 2025 at 06:04:44 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+t^{3} y^{\prime }+3 t^{2} y&={\mathrm e}^{t} \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 16

Order:=6; 
dsolve([diff(y(t),t$2)+t^3*diff(y(t),t)+3*t^2*y(t)=exp(t),y(0) = 0, D(y)(0) = 0],y(t),type='series',t=0);
 

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

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 32

AsymptoticDSolveValue[{D[y[t],{t,2}]+t^3*D[y[t],t]+3*t^2*y[t]==Exp[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],{t,0,"6"-1}]
 

\[ y(t)\to \frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2} \]