60.3.406 problem 1412

Internal problem ID [11416]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1412
Date solved : Monday, January 27, 2025 at 11:20:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 23

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

\[ y = \sinh \left (x \left (-1+\ln \left (x \right )\right )\right ) c_{1} +\cosh \left (x \left (-1+\ln \left (x \right )\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 29

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

\[ y(x)\to c_1 \cosh (x (\log (x)-1))+i c_2 \sinh (x (\log (x)-1)) \]