50.14.16 problem 2(h)

Internal problem ID [8054]
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)
Date solved : Monday, January 27, 2025 at 03:40:21 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }&=\ln \left (x \right ) \end{align*}

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: 0.271 (sec). Leaf size: 47

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)
 

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

✓ Solution by Mathematica

Time used: 0.094 (sec). Leaf size: 66

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

\[ y(x)\to \frac {1}{4} \left (e^{2 x} \operatorname {ExpIntegralEi}(-2 x)-\operatorname {ExpIntegralEi}(-2) e^{2 x}+2 x+2 e^{2 x-3}-2 x \log (x)-\log (-x)+i \pi +4 e-\frac {2}{e}-2\right ) \]