70.18.5 problem 16

Internal problem ID [19002]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.2 (Properties of the Laplace transform). Problems at page 309
Problem number : 16
Date solved : Thursday, October 02, 2025 at 03:36:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+2 y&=t^{2} {\mathrm e}^{t}+7 \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 25

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+2*y(t) = t^2*exp(t)+7; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {7}{2}+\frac {\left (2 t^{2}-\cos \left (t \right )+7 \sin \left (t \right )-4\right ) {\mathrm e}^{t}}{2} \]

✓ Mathematica. Time used: 0.224 (sec). Leaf size: 125

ode=D[y[t],{t,2}]-2*D[y[t],t]+2*y[t]==t^2*Exp[t]+7; 
ic={y[0]==1,Derivative[1][y][0] == 1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -e^t \left (\sin (t) \int _1^0\cos (K[1]) \left (K[1]^2+7 e^{-K[1]}\right )dK[1]-\sin (t) \int _1^t\cos (K[1]) \left (K[1]^2+7 e^{-K[1]}\right )dK[1]+\cos (t) \left (-\int _1^t\left (-K[2]^2-7 e^{-K[2]}\right ) \sin (K[2])dK[2]+\int _1^0\left (-K[2]^2-7 e^{-K[2]}\right ) \sin (K[2])dK[2]-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.157 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*exp(t) + 2*y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 7,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (t^{2} + \frac {7 \sin {\left (t \right )}}{2} - \frac {\cos {\left (t \right )}}{2} - 2\right ) e^{t} + \frac {7}{2} \]