7.10.8 problem 8

Internal problem ID [2409]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.4, The method of variation of parameters. Page 154
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:35:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=f \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 39

ode:=diff(diff(y(t),t),t)-y(t) = f(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-t} \int _{0}^{t}{\mathrm e}^{\textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1}}{2}+\frac {{\mathrm e}^{t} \int _{0}^{t}{\mathrm e}^{-\textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1}}{2} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 103

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

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

✓ Sympy. Time used: 0.436 (sec). Leaf size: 49

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

\[ y{\left (t \right )} = \left (\frac {\int f{\left (t \right )} e^{- t}\, dt}{2} - \frac {\int \limits ^{0} f{\left (t \right )} e^{- t}\, dt}{2}\right ) e^{t} + \left (- \frac {\int f{\left (t \right )} e^{t}\, dt}{2} + \frac {\int \limits ^{0} f{\left (t \right )} e^{t}\, dt}{2}\right ) e^{- t} \]