68.11.44 problem 56

Internal problem ID [17581]
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 : Thursday, October 02, 2025 at 02:25:42 PM
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*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 2.421 (sec). Leaf size: 133

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

\begin{align*} 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 \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 22

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

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