64.11.39 problem 39

Internal problem ID [13418]
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 : Monday, March 31, 2025 at 07:53:45 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,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: 0.219 (sec). Leaf size: 260

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]
 

\[ y(x)\to \frac {1}{80} e^{-x} \left (80 \int _1^x\frac {1}{12} e^{K[1]} \left (e^{K[1]} (3 K[1]+2)-\sin (K[1])\right )dK[1]-80 e^{3 x} \int _1^0\frac {1}{3} e^{-2 K[2]} \left (\sin (K[2])-e^{K[2]} (3 K[2]+2)\right )dK[2]+80 e^{3 x} \int _1^x\frac {1}{3} e^{-2 K[2]} \left (\sin (K[2])-e^{K[2]} (3 K[2]+2)\right )dK[2]-80 e^{4 x} \int _1^0\frac {1}{4} e^{-3 K[3]} \left (e^{K[3]} (3 K[3]+2)-\sin (K[3])\right )dK[3]+80 e^{4 x} \int _1^x\frac {1}{4} e^{-3 K[3]} \left (e^{K[3]} (3 K[3]+2)-\sin (K[3])\right )dK[3]-80 \int _1^0\frac {1}{12} e^{K[1]} \left (e^{K[1]} (3 K[1]+2)-\sin (K[1])\right )dK[1]+66 e^{3 x}-33 e^{4 x}+33\right ) \]

✓ Sympy. Time used: 0.279 (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)
 

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