78.16.18 problem 18

Internal problem ID [18397]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 23. Operator Methods for Finding Particular Solutions. Problems at page 169
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 11:49:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 35

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

\[ y = \frac {\left ({\mathrm e}^{4 x}+\left (3 c_{1} x +3 c_{2} \right ) {\mathrm e}^{x}+6 x^{2}+24 x +36\right ) {\mathrm e}^{-2 x}}{3} \]

✓ Solution by Mathematica

Time used: 0.190 (sec). Leaf size: 42

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

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