74.14.29 problem 29

Internal problem ID [16363]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 29
Date solved : Monday, March 31, 2025 at 02:51:12 PM
CAS classification : [[_high_order, _missing_y]]

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

With initial conditions

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 15

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+diff(diff(y(t),t),t) = cos(t); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 60.046 (sec). Leaf size: 192

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

\[ y(t)\to -t \int _1^0\frac {1}{2} \left (\cos ^3(K[2])+\cos (K[2])-2 \sin (K[2]) \int _1^0\cos ^2(K[1])dK[1]+2 \sin (K[2]) \int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]+\int _1^t\int _1^{K[3]}\frac {1}{2} \left (\cos ^3(K[2])+\cos (K[2])-2 \sin (K[2]) \int _1^0\cos ^2(K[1])dK[1]+2 \sin (K[2]) \int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]dK[3]-\int _1^0\int _1^{K[3]}\frac {1}{2} \left (\cos ^3(K[2])+\cos (K[2])-2 \sin (K[2]) \int _1^0\cos ^2(K[1])dK[1]+2 \sin (K[2]) \int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]dK[3] \]

✓ Sympy. Time used: 0.141 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-cos(t) + Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0, Subs(Derivative(y(t), (t, 2)), t, 0): 1, Subs(Derivative(y(t), (t, 3)), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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