60.3.152 problem 1156

Internal problem ID [11162]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1156
Date solved : Tuesday, January 28, 2025 at 05:41:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}-x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.156 (sec). Leaf size: 75

dsolve(x^2*diff(diff(y(x),x),x)+y(x)/ln(x)-x*exp(x)*(2+x*ln(x))=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.249 (sec). Leaf size: 76

DSolve[-(E^x*x*(2 + x*Log[x])) + y[x]/Log[x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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