76.17.14 problem 23

Internal problem ID [17629]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 23
Date solved : Monday, March 31, 2025 at 04:23:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 31

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(t), t) - (-t**2*exp(2*t) + t*Derivative(y(t), (t, 2)) + y(t))/(t + 1) cannot be solved by the factorable group method