64.11.44 problem 44

Internal problem ID [13494]
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 : 44
Date solved : Tuesday, January 28, 2025 at 05:46:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=x^{4} {\mathrm e}^{x}+x^{3} {\mathrm e}^{2 x}+x^{2} {\mathrm e}^{3 x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 62

dsolve(diff(y(x),x$2)-6*diff(y(x),x)+9*y(x)=x^4*exp(x)+x^3*exp(2*x)+x^2*exp(3*x),y(x), singsol=all)
 

\[ y = \left (x^{3}+6 x^{2}+18 x +24\right ) {\mathrm e}^{2 x}+\frac {\left (x^{4}+12 c_{1} x +12 c_{2} \right ) {\mathrm e}^{3 x}}{12}+\frac {\left (x^{4}+4 x^{3}+9 x^{2}+12 x +\frac {15}{2}\right ) {\mathrm e}^{x}}{4} \]

✓ Solution by Mathematica

Time used: 1.653 (sec). Leaf size: 91

DSolve[D[y[x],{x,2}]-6*D[y[x],x]+9*y[x]==x^4*Exp[x]+x^3*Exp[2*x]+x^2*Exp[3*x],y[x],x,IncludeSingularSolutions -> True]
 

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