84.37.7 problem 26.7

Internal problem ID [22349]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Solved problems. Page 159
Problem number : 26.7
Date solved : Thursday, October 02, 2025 at 08:37:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)-diff(y(t),t)-2*y(t) = 4*t^2; 
ic:=[y(0) = 1, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 \,{\mathrm e}^{2 t}-3+2 t +2 \,{\mathrm e}^{-t}-2 t^{2} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 29

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

\begin{align*} y(t)&\to -2 t^2+2 t+2 e^{-t}+2 e^{2 t}-3 \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 24

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

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