58.11.39 problem 39

Internal problem ID [14770]
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 : 39
Date solved : Thursday, October 02, 2025 at 09:51:00 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-4 y^{\prime \prime }+y^{\prime }+6 y&=3 x \,{\mathrm e}^{x}+2 \,{\mathrm e}^{x}-\sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {33}{40}} \\ y^{\prime }\left (0\right )&=0 \\ y^{\prime \prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 30

ode:=diff(diff(diff(y(x),x),x),x)-4*diff(diff(y(x),x),x)+diff(y(x),x)+6*y(x) = 3*x*exp(x)+2*exp(x)-sin(x); 
ic:=[y(0) = 33/40, D(y)(0) = 0, (D@@2)(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {\sin \left (x \right )}{10}-\frac {31 \,{\mathrm e}^{2 x}}{40}+\frac {7 \,{\mathrm e}^{-x}}{20}+\frac {\left (5+3 x \right ) {\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 38

ode=D[y[x],{x,3}]-4*D[y[x],{x,2}]+D[y[x],x]+6*y[x]==3*x*Exp[x]+2*Exp[x]-Sin[x]; 
ic={y[0]==33/40,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{40} \left (10 e^x (3 x+5)+14 e^{-x}-31 e^{2 x}-4 \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*exp(x) + 6*y(x) - 2*exp(x) + sin(x) + Derivative(y(x), x) - 4*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 33/40, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (3 x + 5\right ) e^{x}}{4} - \frac {31 e^{2 x}}{40} - \frac {\sin {\left (x \right )}}{10} + \frac {7 e^{- x}}{20} \]