14.12.13 problem 13

Internal problem ID [2623]
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 : 13
Date solved : Sunday, March 30, 2025 at 12:11:49 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (1-t \right ) y^{\prime \prime }+t y^{\prime }+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: 0.004 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 33

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

\[ y(t)\to \frac {7 t^5}{120}+\frac {t^4}{24}-\frac {t^3}{6}-\frac {t^2}{2}+1 \]

✓ Sympy. Time used: 0.814 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) + (1 - t)*Derivative(y(t), (t, 2)) + y(t),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)
 

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