Internal problem ID [5639]
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: 3(f).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+4 y^{\prime }+4 y-x \ln \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 42
dsolve(diff(y(x),x$2)+4*diff(y(x),x)+4*y(x)=x*ln(x),y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{-2 x}+{\mathrm e}^{-2 x} x c_{1}-\frac {{\mathrm e}^{-2 x} \left (x +1\right ) \expIntegral \left (1, -2 x \right )}{4}-\frac {3}{8}+\frac {\left (2 x -2\right ) \ln \relax (x )}{8} \]
✓ Solution by Mathematica
Time used: 0.039 (sec). Leaf size: 48
DSolve[y''[x]+4*y'[x]+4*y[x]==x*Log[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{8} e^{-2 x} \left (2 (x+1) \text {ExpIntegralEi}(2 x)+e^{2 x} (2 (x-1) \log (x)-3)+8 (c_2 x+c_1)\right ) \\ \end{align*}