74.11.44 problem 56

Internal problem ID [16294]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 56
Date solved : Tuesday, January 28, 2025 at 09:00:50 AM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&={\frac {3}{2}} \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 23

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

\[ y = {\mathrm e}^{2 t}-\frac {t^{2}}{2}-\frac {t^{3}}{3}-\frac {t}{2}+2 \]

✓ Solution by Mathematica

Time used: 4.307 (sec). Leaf size: 133

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

\[ y(t)\to \int _1^t\frac {1}{2} e^{2 K[2]} \left (-2 \int _1^02 e^{-2 K[1]} K[1]^2dK[1]+2 \int _1^{K[2]}2 e^{-2 K[1]} K[1]^2dK[1]+3\right )dK[2]-\int _1^0\frac {1}{2} e^{2 K[2]} \left (-2 \int _1^02 e^{-2 K[1]} K[1]^2dK[1]+2 \int _1^{K[2]}2 e^{-2 K[1]} K[1]^2dK[1]+3\right )dK[2]+3 \]