68.11.43 problem 55

Internal problem ID [17580]
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 : 55
Date solved : Thursday, October 02, 2025 at 02:25:41 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-y^{\prime }&=-3 t -4 \,{\mathrm e}^{2 t} t^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {7}{2}} \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 33

ode:=diff(diff(y(t),t),t)-diff(y(t),t) = -3*t-4*t^2*exp(2*t); 
ic:=[y(0) = -7/2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {3}{2}+\left (-2 t^{2}+6 t -7\right ) {\mathrm e}^{2 t}+\frac {3 t^{2}}{2}+3 t +5 \,{\mathrm e}^{t} \]

✓ Mathematica. Time used: 3.849 (sec). Leaf size: 164

ode=D[y[t],{t,2}]-D[y[t],t]==-3*t-4*t^2*Exp[2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==8/9}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \int _1^t\frac {1}{9} e^{K[2]} \left (-9 \int _1^0\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+9 \int _1^{K[2]}\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+8\right )dK[2]-\int _1^0\frac {1}{9} e^{K[2]} \left (-9 \int _1^0\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+9 \int _1^{K[2]}\left (-4 e^{K[1]} K[1]^2-3 e^{-K[1]} K[1]\right )dK[1]+8\right )dK[2] \end{align*}

✓ Sympy. Time used: 0.173 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*t**2*exp(2*t) + 3*t - Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): -7/2, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {3 t^{2}}{2} + 3 t + \left (- 2 t^{2} + 6 t - 7\right ) e^{2 t} + 5 e^{t} - \frac {3}{2} \]