7.12.16 problem 15

Internal problem ID [2428]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 05:35:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+t y^{\prime }+{\mathrm e}^{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 )&=0 \\ \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 18

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

\[ y = 1-\frac {1}{2} t^{2}-\frac {1}{6} t^{3}+\frac {1}{12} t^{4}+\frac {1}{20} t^{5}+\operatorname {O}\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 33

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

\[ y(t)\to \frac {t^5}{20}+\frac {t^4}{12}-\frac {t^3}{6}-\frac {t^2}{2}+1 \]

✓ Sympy. Time used: 0.287 (sec). Leaf size: 54

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4} e^{2 t}}{24} + \frac {t^{4} e^{t}}{12} - \frac {t^{2} e^{t}}{2} + 1\right ) + C_{1} t \left (- \frac {t^{2} e^{t}}{6} - \frac {t^{2}}{6} + 1\right ) + O\left (t^{6}\right ) \]