76.15.17 problem 18

Internal problem ID [17587]
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 : 18
Date solved : Monday, March 31, 2025 at 04:18:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=t^{2}+3 \,{\mathrm e}^{t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)+4*y(t) = t^2+3*exp(t); 
ic:=y(0) = 0, D(y)(0) = 2; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {7 \sin \left (2 t \right )}{10}-\frac {19 \cos \left (2 t \right )}{40}+\frac {t^{2}}{4}-\frac {1}{8}+\frac {3 \,{\mathrm e}^{t}}{5} \]

✓ Mathematica. Time used: 0.195 (sec). Leaf size: 33

ode=D[y[t],{t,2}]+4*y[t]==t^2+3*Exp[t]; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{40} \left (10 t^2+24 e^t+28 \sin (2 t)-19 \cos (2 t)-5\right ) \]

✓ Sympy. Time used: 0.112 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 + 4*y(t) - 3*exp(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2}}{4} + \frac {3 e^{t}}{5} + \frac {7 \sin {\left (2 t \right )}}{10} - \frac {19 \cos {\left (2 t \right )}}{40} - \frac {1}{8} \]