68.14.30 problem 30

Internal problem ID [17722]
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 : 30
Date solved : Thursday, October 02, 2025 at 02:27:22 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }&=t \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: 0.032 (sec). Leaf size: 20

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

\[ y = \frac {t^{3}}{6}-\cos \left (t \right )+\sin \left (t \right )-t +1 \]

✓ Mathematica. Time used: 60.016 (sec). Leaf size: 81

ode=D[y[t],{t,4}]+D[y[t],{t,2}]==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]
 

\begin{align*} y(t)&\to -t \int _1^0(\cos (K[1])+K[1]-\sin (K[1]))dK[1]+\int _1^t\int _1^{K[2]}(\cos (K[1])+K[1]-\sin (K[1]))dK[1]dK[2]-\int _1^0\int _1^{K[2]}(\cos (K[1])+K[1]-\sin (K[1]))dK[1]dK[2] \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 17

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

\[ y{\left (t \right )} = \frac {t^{3}}{6} - t + \sin {\left (t \right )} - \cos {\left (t \right )} + 1 \]