64.12.18 problem 18

Internal problem ID [13522]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 05:51:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)-2*diff(y(x),x)+y(x)=x*ln(x),y(x), singsol=all)
 

\[ y = -{\mathrm e}^{x} \left (x -2\right ) \operatorname {Ei}_{1}\left (x \right )+\left (c_{1} x +c_{2} \right ) {\mathrm e}^{x}+3+\left (x +2\right ) \ln \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.090 (sec). Leaf size: 59

DSolve[D[y[x],{x,2}]-2*D[y[x],x]+y[x]==x*Log[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x \left (\int _1^x-e^{-K[1]} K[1]^2 \log (K[1])dK[1]+x \int _1^xe^{-K[2]} K[2] \log (K[2])dK[2]+c_2 x+c_1\right ) \]