68.11.33 problem 45

Internal problem ID [17570]
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 : 45
Date solved : Thursday, October 02, 2025 at 02:25:33 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=t^{2}+{\mathrm e}^{t}+\sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t) = t^2+exp(t)+sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {t^{4}}{12}-\sin \left (t \right )+{\mathrm e}^{t}+c_1 t +c_2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 39

ode=D[y[t],{t,2}]==t^2+Exp[t]+Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \int _1^t\int _1^{K[2]}\left (K[1]^2+e^{K[1]}+\sin (K[1])\right )dK[1]dK[2]+c_2 t+c_1 \end{align*}

✓ Sympy. Time used: 0.049 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = C_{1} + C_{2} t + \frac {t^{4}}{12} + e^{t} - \sin {\left (t \right )} \]