64.11.39 problem 39

Internal problem ID [13489]
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 : Tuesday, January 28, 2025 at 05:45:45 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*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.219 (sec). Leaf size: 260

DSolve[{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],{y[0]==33/40,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

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