14.16 problem 2(h)

Internal problem ID [5633]

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: 2(h).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _missing_y]]

Solve \begin {gather*} \boxed {y^{\prime \prime }-2 y^{\prime }-\ln \relax (x )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (1) = {\mathrm e}, y^{\prime }\relax (1) = {\mathrm e}^{-1}] \end {align*}

Solution by Maple

Time used: 0.235 (sec). Leaf size: 44

dsolve([diff(y(x),x$2)-2*diff(y(x),x)=ln(x),y(1) = exp(1), D(y)(1) = 1/exp(1)],y(x), singsol=all)
 

\[ y \relax (x ) = \frac {\left (\int _{1}^{x}\left (\left (\expIntegral \left (1, 2\right )-\expIntegral \left (1, 2 \textit {\_z1} \right )\right ) {\mathrm e}^{2 \textit {\_z1}}-\ln \left (\textit {\_z1} \right )+2 \,{\mathrm e}^{2 \textit {\_z1} -3}\right )d \textit {\_z1} \right )}{2}+{\mathrm e} \]

Solution by Mathematica

Time used: 0.136 (sec). Leaf size: 66

DSolve[{y''[x]-2*y'[x]==Log[x],{y[1]==Exp[1],y'[1]==1/Exp[1]}},y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} \left (e^{2 x} \text {Ei}(-2 x)-\text {Ei}(-2) e^{2 x}+2 x+2 e^{2 x-3}-2 x \log (x)-\log (-x)+i \pi +4 e-\frac {2}{e}-2\right ) \\ \end{align*}