74.14.26 problem 26

Internal problem ID [16360]
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 : 26
Date solved : Monday, March 31, 2025 at 02:51:07 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }&=3 t^{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 28

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

\[ y = 6 \,{\mathrm e}^{t}-3 t^{2}-t^{3}-\frac {t^{4}}{4}-6 t -6 \]

✓ Mathematica. Time used: 11.592 (sec). Leaf size: 130

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

\[ y(t)\to \int _1^t\left (e^{K[2]} \left (\int _1^{K[2]}3 e^{-K[1]} K[1]^2dK[1]-\int _1^03 e^{-K[1]} K[1]^2dK[1]\right )-K[2]^3\right )dK[2]-\int _1^0\left (e^{K[2]} \left (\int _1^{K[2]}3 e^{-K[1]} K[1]^2dK[1]-\int _1^03 e^{-K[1]} K[1]^2dK[1]\right )-K[2]^3\right )dK[2] \]

✓ Sympy. Time used: 0.105 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = - \frac {t^{4}}{4} - t^{3} - 3 t^{2} - 6 t + 6 e^{t} - 6 \]