7.12.7 problem 7

Internal problem ID [2419]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 05:35:37 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-t^{3} y&=0 \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 )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 10

Order:=6; 
ode:=diff(diff(y(t),t),t)-t^3*y(t) = 0; 
ic:=[y(0) = 0, D(y)(0) = -2]; 
dsolve([ode,op(ic)],y(t),type='series',t=0);
 

\[ y = \left (-2\right ) t +\operatorname {O}\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 6

ode=D[y[t],{t,2}]-t^3*y[t]==0; 
ic={y[0]==0,Derivative[1][y][0] ==-2}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to -2 t \]

✓ Sympy. Time used: 0.184 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**3*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): -2} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{5}}{20} + 1\right ) + C_{1} t \left (\frac {t^{5}}{30} + 1\right ) + O\left (t^{6}\right ) \]