75.16.54 problem 527

Internal problem ID [17027]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Trial and error method. Exercises page 132
Problem number : 527
Date solved : Tuesday, January 28, 2025 at 09:46:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\left (x^{2}+x \right ) {\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 43

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

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

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 65

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

\[ y(x)\to \frac {1}{9} e^{-x/2} \left (e^{3 x/2} \left (3 x^2-3 x+1\right )+9 c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+9 c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]