50.14.27 problem 4(c)

Internal problem ID [8065]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 4(c)
Date solved : Monday, January 27, 2025 at 03:41:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=x^{2}+2 x +2 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 51

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

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