64.12.26 problem 26

Internal problem ID [13530]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 26
Date solved : Tuesday, January 28, 2025 at 05:52:02 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }-y^{\prime }+3 y&=x^{2} {\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 31

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

\[ y = c_{2} {\mathrm e}^{-x}+c_{3} {\mathrm e}^{3 x}-\frac {\left (x^{3}+\frac {3}{2} x -12 c_{1} \right ) {\mathrm e}^{x}}{12} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 92

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

\[ y(x)\to e^{-x} \int _1^x\frac {1}{8} e^{2 K[1]} K[1]^2dK[1]+e^{3 x} \int _1^x\frac {1}{8} e^{-2 K[2]} K[2]^2dK[2]-\frac {1}{12} e^x x^3+c_1 e^{-x}+c_2 e^x+c_3 e^{3 x} \]