64.11.19 problem 19

Internal problem ID [13469]
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 : 19
Date solved : Tuesday, January 28, 2025 at 05:44:40 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=x \,{\mathrm e}^{x}-4 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{4 x} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 42

dsolve(diff(y(x),x$3)-6*diff(y(x),x$2)+11*diff(y(x),x)-6*y(x)=x*exp(x)-4*exp(2*x)+6*exp(4*x),y(x), singsol=all)
 

\[ y = \left (4 x +c_{2} \right ) {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{3 x}+{\mathrm e}^{4 x}+\frac {\left (2 x^{2}+8 c_{1} +6 x +7\right ) {\mathrm e}^{x}}{8} \]

✓ Solution by Mathematica

Time used: 0.262 (sec). Leaf size: 121

DSolve[D[y[x],{x,3}]-6*D[y[x],{x,2}]+11*D[y[x],x]-6*y[x]==x*Exp[x]-4*Exp[2*x]+6*Exp[4*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{4} e^x \left (4 e^x \int _1^x\left (-e^{-K[1]} K[1]-6 e^{2 K[1]}+4\right )dK[1]+4 e^{2 x} \int _1^x\frac {1}{2} e^{-2 K[2]} \left (K[2]-4 e^{K[2]}+6 e^{3 K[2]}\right )dK[2]+x^2-8 e^x+4 e^{3 x}+4 c_2 e^x+4 c_3 e^{2 x}+4 c_1\right ) \]