7.12.15 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 : Tuesday, September 30, 2025 at 05:35:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

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

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

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

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

✓ Sympy. Time used: 0.221 (sec). Leaf size: 37

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4}}{24} - \frac {t^{3}}{6} + 1\right ) + C_{1} t \left (- \frac {t^{3}}{8} + \frac {t^{2}}{6} - \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]