56.1.56 problem 56

Internal problem ID [8768]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 56
Date solved : Wednesday, March 05, 2025 at 06:46:17 AM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} y^{\prime \prime }&=f \left (t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

ode:=diff(diff(y(t),t),t) = f(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \int \left (\int f \left (t \right )d t \right )d t +c_{1} t +c_{2} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 30

ode=D[y[t],{t,2}]==f[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \int _1^t\int _1^{K[2]}f(K[1])dK[1]dK[2]+c_2 t+c_1 \]

✓ Sympy. Time used: 0.312 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = C_{1} + t \left (C_{2} + \int f{\left (t \right )}\, dt\right ) - \int t f{\left (t \right )}\, dt \]