83.15.10 problem 4 (j)

Internal problem ID [22034]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 4 (j)
Date solved : Thursday, October 02, 2025 at 08:21:51 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }-2 y^{\prime }+y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=1 \\ y^{\prime \prime }\left (0\right )&=-3 \\ y^{\prime \prime \prime }\left (0\right )&=-5 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 43

ode:=diff(diff(diff(diff(y(t),t),t),t),t)-2*diff(diff(diff(y(t),t),t),t)-2*diff(y(t),t)+y(t) = 0; 
ic:=[y(0) = 3, D(y)(0) = 1, (D@@2)(y)(0) = -3, (D@@3)(y)(0) = -5]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-2 \textit {\_Z} +1\right )}{\sum }\left (6 \underline {\hspace {1.25 ex}}\alpha ^{3}-10 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha -15\right ) {\mathrm e}^{\underline {\hspace {1.25 ex}}\alpha t}\right )}{6} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 7369

ode=D[y[t],{t,4}]-2*D[y[t],{t,3}]-2*D[y[t],t]+y[t]==0; 
ic={y[0]==3,Derivative[1][y][0] ==1,Derivative[2][y][0] ==-3,Derivative[3][y][0] ==-5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

Too large to display

✓ Sympy. Time used: 0.484 (sec). Leaf size: 207

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) - 2*Derivative(y(t), t) - 2*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {y(0): 3, Subs(Derivative(y(t), t), t, 0): 1, Subs(Derivative(y(t), (t, 2)), t, 0): -3, Subs(Derivative(y(t), (t, 3)), t, 0): -5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{3} + \frac {7 \sqrt {2} \sqrt [4]{3}}{6}\right ) e^{\frac {t \left (1 - \sqrt {3}\right )}{2}} \sin {\left (\frac {\sqrt {2} \sqrt [4]{3} t}{2} \right )} + \left (\frac {3}{2} - \frac {\sqrt {3}}{2}\right ) e^{\frac {t \left (1 - \sqrt {3}\right )}{2}} \cos {\left (\frac {\sqrt {2} \sqrt [4]{3} t}{2} \right )} + \left (- \frac {7 \sqrt {2} \sqrt [4]{3}}{12} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{6} + \frac {\sqrt {3}}{4} + \frac {3}{4}\right ) e^{\frac {t \left (1 + \sqrt {3} + \sqrt {2} \sqrt [4]{3}\right )}{2}} + \left (\frac {\sqrt {3}}{4} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}}}{6} + \frac {3}{4} + \frac {7 \sqrt {2} \sqrt [4]{3}}{12}\right ) e^{\frac {t \left (- \sqrt {2} \sqrt [4]{3} + 1 + \sqrt {3}\right )}{2}} \]