74.12.51 problem 60

Internal problem ID [16279]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 60
Date solved : Thursday, March 13, 2025 at 08:09:31 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=f \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.349 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 106

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

\[ y(t)\to \sin (2 t) \left (-\int _1^0\frac {1}{2} \cos (2 K[2]) f(K[2])dK[2]\right )+\sin (2 t) \int _1^t\frac {1}{2} \cos (2 K[2]) f(K[2])dK[2]-\cos (2 t) \int _1^0-\cos (K[1]) f(K[1]) \sin (K[1])dK[1]+\cos (2 t) \int _1^t-\cos (K[1]) f(K[1]) \sin (K[1])dK[1]+\sin (2 t) \]

✓ Sympy. Time used: 1.228 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
y = Function("y") 
f = Function("f") 
ode = Eq(-f(t) + 4*y(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)
 

\[ y{\left (t \right )} = \left (- \frac {\int f{\left (t \right )} \sin {\left (2 t \right )}\, dt}{2} + \frac {\int \limits ^{0} f{\left (t \right )} \sin {\left (2 t \right )}\, dt}{2}\right ) \cos {\left (2 t \right )} + \left (\frac {\int f{\left (t \right )} \cos {\left (2 t \right )}\, dt}{2} - \frac {\int \limits ^{0} f{\left (t \right )} \cos {\left (2 t \right )}\, dt}{2} + 1\right ) \sin {\left (2 t \right )} \]