74.14.26 problem 26

Internal problem ID [16431]
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 : Tuesday, January 28, 2025 at 09:07:52 AM
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*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

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

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

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