12.19.71 problem section 9.3, problem 71

Internal problem ID [2218]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 71
Date solved : Tuesday, March 04, 2025 at 01:51:54 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime \prime }-3 y^{\prime }-y&={\mathrm e}^{-\frac {x}{2}} \left (2-3 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=15\\ y^{\prime \prime }\left (0\right )&=-17 \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 22

ode:=4*diff(diff(diff(y(x),x),x),x)-3*diff(y(x),x)-y(x) = exp(-1/2*x)*(2-3*x); 
ic:=y(0) = -1, D(y)(0) = 15, (D@@2)(y)(0) = -17; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (x^{3}+192 x \right ) {\mathrm e}^{-\frac {x}{2}}}{12}-{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 35

ode=4*D[y[x],{x,3}]-0*D[y[x],{x,2}]-3*D[y[x],x]-1*y[x]==Exp[-x/2]*(2-3*x); 
ic={y[0]==2,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{36} e^{-x/2} \left (3 x^3+24 x+8 e^{3 x/2}+64\right ) \]

✓ Sympy. Time used: 0.369 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x - 2)*exp(-x/2) - y(x) - 3*Derivative(y(x), x) + 4*Derivative(y(x), (x, 3)),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 15, Subs(Derivative(y(x), (x, 2)), x, 0): -17} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x \left (\frac {x^{2}}{12} + 16\right ) e^{- \frac {x}{2}} - e^{x} \]