87.22.16 problem 16

Internal problem ID [23761]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 16
Date solved : Thursday, October 02, 2025 at 09:44:57 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-y&=-1 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-2 \\ y^{\prime \prime }\left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 22

ode:=diff(diff(diff(y(t),t),t),t)-y(t) = -1; 
ic:=[y(0) = 1, D(y)(0) = -2, (D@@2)(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {4 \sqrt {3}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right )}{3}+1 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 32

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

\begin{align*} y(t)&\to 1-\frac {4 e^{-t/2} \sin \left (\frac {\sqrt {3} t}{2}\right )}{\sqrt {3}} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) + Derivative(y(t), (t, 2)) + 1,0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -2, Subs(Derivative(y(t), (t, 2)), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

ValueError : Couldnt solve for initial conditions