Internal problem ID [9485]
Internal file name [OUTPUT/8425_Monday_June_06_2022_03_02_23_AM_74390002/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1156.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {x^{2} y^{\prime \prime }+\frac {y}{\ln \left (x \right )}=x \,{\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] trying symmetries linear in x and y(x) -> Try solving first the homogeneous part of the ODE trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful Change of variables used: [x = exp(t)] Linear ODE actually solved: u(t)-t*diff(u(t),t)+t*diff(diff(u(t),t),t) = 0 <- change of variables successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.094 (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)
\[ y \left (x \right ) = -\ln \left (x \right )^{3} \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right ) {\mathrm e}^{x}-\ln \left (x \right )^{2} {\mathrm e}^{x} x -\ln \left (x \right ) \operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right ) c_{1} +\left (\int \frac {\left (\operatorname {expIntegral}_{1}\left (-\ln \left (x \right )\right ) \ln \left (x \right )+x \right ) {\mathrm e}^{x} \left (2+x \ln \left (x \right )\right )}{x}d x \right ) \ln \left (x \right )+\ln \left (x \right ) c_{2} -c_{1} x \]
✓ Solution by Mathematica
Time used: 0.127 (sec). Leaf size: 27
DSolve[-(E^x*x*(2 + x*Log[x])) + y[x]/Log[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \operatorname {LogIntegral}(x) \log (x)+c_2 (-x)+\left (e^x+c_1\right ) \log (x) \]