70.15.18 problem 19

Internal problem ID [18946]
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 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 19
Date solved : Thursday, October 02, 2025 at 03:33:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&={\mathrm e}^{t} t +4 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 18

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+y(t) = t*exp(t)+4; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = 4+\frac {\left (t^{3}+24 t -18\right ) {\mathrm e}^{t}}{6} \]

✓ Mathematica. Time used: 0.15 (sec). Leaf size: 83

ode=D[y[t],{t,2}]-2*D[y[t],t]+y[t]==t*Exp[t]+4; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^t \int _1^0\left (-K[1]-4 e^{-K[1]}\right ) K[1]dK[1]+e^t \int _1^t\left (-K[1]-4 e^{-K[1]}\right ) K[1]dK[1]+\frac {e^t t^3}{2}+4 e^t t-4 t+e^t \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*exp(t) + y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 4,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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