Internal problem ID [6386]
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]]
\[ \boxed {y^{\prime \prime }-2 y^{\prime }=\ln \left (x \right )} \] With initial conditions \begin {align*} [y \left (1\right ) = {\mathrm e}, y^{\prime }\left (1\right ) = {\mathrm e}^{-1}] \end {align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 42
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)
\[ y \left (x \right ) = \frac {\left (\int _{1}^{x}\left (-{\mathrm e}^{2 \textit {\_z1}} \operatorname {Ei}_{1}\left (2 \textit {\_z1} \right )+{\mathrm e}^{2 \textit {\_z1}} \operatorname {Ei}_{1}\left (2\right )-\ln \left (\textit {\_z1} \right )+2 \,{\mathrm e}^{2 \textit {\_z1} -3}\right )d \textit {\_z1} \right )}{2}+{\mathrm e} \]
✓ Solution by Mathematica
Time used: 0.101 (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]
\[ y(x)\to \frac {1}{4} \left (e^{2 x} \operatorname {ExpIntegralEi}(-2 x)-\operatorname {ExpIntegralEi}(-2) e^{2 x}+2 x+2 e^{2 x-3}-2 x \log (x)-\log (-x)+i \pi +4 e-\frac {2}{e}-2\right ) \]