77.1.116 problem 142 (page 205)

Internal problem ID [18006]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 142 (page 205)
Date solved : Tuesday, January 28, 2025 at 08:28:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.465 (sec). Leaf size: 75

dsolve(diff(y(x),x$2)+1/(x^2*ln(x))*y(x)=exp(x)*(2/x+ln(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.108 (sec). Leaf size: 27

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

\[ y(x)\to c_2 \operatorname {LogIntegral}(x) \log (x)+c_2 (-x)+\left (e^x+c_1\right ) \log (x) \]