Internal problem ID [5644]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+y^{\prime }+y-x^{2}-2 x -2=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 34
dsolve([diff(y(x),x$2)+diff(y(x),x)+y(x)=x^2+2*x+2,x^2],y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1}+x^{2} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 46
DSolve[y''[x]+y'[x]+y[x]==x^2+2*x+2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2+e^{-x/2} \left (c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \\ \end{align*}