14.12.6 problem 6

Internal problem ID [2616]
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 : 6
Date solved : Monday, January 27, 2025 at 06:04:38 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+t^{2} y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

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

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

\[ y = 2-t -\frac {1}{6} t^{4}+\frac {1}{20} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 22

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

\[ y(t)\to \frac {t^5}{20}-\frac {t^4}{6}-t+2 \]