71.12.1 problem 1

Internal problem ID [14446]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.5, page 221
Problem number : 1
Date solved : Monday, March 31, 2025 at 12:27:23 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }-4 y^{\prime }+12 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 23

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)-4*diff(y(x),x)+12*y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 5, (D@@2)(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -{\mathrm e}^{-2 x}-{\mathrm e}^{3 x}+3 \,{\mathrm e}^{2 x} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 26

ode=D[y[x],{x,3}]-3*D[y[x],{x,2}]-4*D[y[x],x]+12*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==5,Derivative[2][y][0] ==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -e^{-2 x} \left (-3 e^{4 x}+e^{5 x}+1\right ) \]

✓ Sympy. Time used: 0.213 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(12*y(x) - 4*Derivative(y(x), x) - 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 5, Subs(Derivative(y(x), (x, 2)), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

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