7.12.6 problem 6

Internal problem ID [2418]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 05:35:36 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*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 16

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

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

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

ode=D[y[t],{t,2}]+t^2*y[t]==0; 
ic={y[0]==2,Derivative[1][y][0] ==-1}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

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

✓ Sympy. Time used: 0.181 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): -1} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (1 - \frac {t^{4}}{12}\right ) + C_{1} t \left (1 - \frac {t^{4}}{20}\right ) + O\left (t^{6}\right ) \]