10.10.22 problem 32

Internal problem ID [1354]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, section 3.6, Variation of Parameters. page 190
Problem number : 32
Date solved : Saturday, March 29, 2025 at 10:53:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-t \right ) y^{\prime \prime }+t y^{\prime }-y&=2 \left (t -1\right ) {\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

ode:=(1-t)*diff(diff(y(t),t),t)+t*diff(y(t),t)-y(t) = 2*(t-1)*exp(-t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = t c_2 +{\mathrm e}^{t} c_1 -2 \,{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (t -1\right ) t +2 \,\operatorname {Ei}_{1}\left (2 t -2\right ) {\mathrm e}^{t -2}+{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.22 (sec). Leaf size: 47

ode=(1-t)*D[y[t],{t,2}]+t*D[y[t],t]-y[t] ==2*(t-1)*Exp[-t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -2 e^{t-2} \operatorname {ExpIntegralEi}(2-2 t)+\frac {2 t \operatorname {ExpIntegralEi}(1-t)}{e}+e^{-t}+c_1 e^t-c_2 t \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), t) + (1 - t)*Derivative(y(t), (t, 2)) - (2*t - 2)*exp(-t) - y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(t), t) - Derivative(y(t), (t, 2)) - 2*exp(-t) - y(t)/t + Derivative(y(t), (t, 2))/t + 2*exp(-t)/t cannot be solved by the factorable group method