58.11.40 problem 40

Internal problem ID [14771]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 40
Date solved : Thursday, October 02, 2025 at 09:51:01 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }-4 y&=8 x^{2}+3-6 \,{\mathrm e}^{2 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=7 \\ y^{\prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 35

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+9*diff(y(x),x)-4*y(x) = 8*x^2+3-6*exp(2*x); 
ic:=[y(0) = 1, D(y)(0) = 7, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -2 x^{2}-9 x +3 \,{\mathrm e}^{2 x}-15+\frac {44 \,{\mathrm e}^{x}}{3}-\frac {5 \,{\mathrm e}^{4 x}}{3}+2 \,{\mathrm e}^{x} x \]

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 42

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+9*D[y[x],x]-4*y[x]==8*x^2+3-6*Exp[2*x]; 
ic={y[0]==1,Derivative[1][y][0] ==7,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -2 x^2-9 x+3 e^{2 x}-\frac {5 e^{4 x}}{3}+e^x \left (2 x+\frac {44}{3}\right )-15 \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**2 - 4*y(x) + 6*exp(2*x) + 9*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) - 3,0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 7, Subs(Derivative(y(x), (x, 2)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - 2 x^{2} - 9 x + \left (2 x + \frac {44}{3}\right ) e^{x} - \frac {5 e^{4 x}}{3} + 3 e^{2 x} - 15 \]