Internal problem ID [8863]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1283.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime } x^{2}+4 x^{2} \ln \relax (x ) y^{\prime }+\left (x^{2} \ln \relax (x )^{2}+2 x -8\right ) y-4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 52
dsolve(4*x^2*diff(diff(y(x),x),x)+4*x^2*ln(x)*diff(y(x),x)+(x^2*ln(x)^2+2*x-8)*y(x)-4*x^2*(exp(x)/(x^x))^(1/2)=0,y(x), singsol=all)
\[ y \relax (x ) = x^{-\frac {x}{2}-1} {\mathrm e}^{\frac {x}{2}} c_{2}+x^{-\frac {x}{2}+2} {\mathrm e}^{\frac {x}{2}} c_{1}+\frac {\sqrt {x^{-x} {\mathrm e}^{x}}\, x^{2} \left (3 \ln \relax (x )-1\right )}{9} \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 78
DSolve[-4*x^2*Sqrt[E^x/x^x] + (-8 + 2*x + x^2*Log[x]^2)*y[x] + 4*x^2*Log[x]*y'[x] + 4*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\sqrt {e^x x^{-x}} x^3+3 \sqrt {e^x x^{-x}} x^3 \log (x)+3 e^{x/2} x^{-x/2} \left (c_2 x^3+3 c_1\right )}{9 x} \\ \end{align*}