Internal problem ID [9152]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1573.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _fully, _exact, _linear]]
Solve \begin {gather*} \boxed {\left ({\mathrm e}^{x}+2 x \right ) y^{\prime \prime \prime \prime }+4 \left ({\mathrm e}^{x}+2\right ) y^{\prime \prime \prime }+6 \,{\mathrm e}^{x} y^{\prime \prime }+4 \,{\mathrm e}^{x} y^{\prime }+y \,{\mathrm e}^{x}-\frac {1}{x^{5}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 65
dsolve((exp(x)+2*x)*diff(diff(diff(diff(y(x),x),x),x),x)+4*(exp(x)+2)*diff(diff(diff(y(x),x),x),x)+6*exp(x)*diff(diff(y(x),x),x)+4*exp(x)*diff(y(x),x)+y(x)*exp(x)-1/x^5=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{4}}{{\mathrm e}^{x}+2 x}+\frac {x c_{3}}{{\mathrm e}^{x}+2 x}+\frac {1}{24 \left ({\mathrm e}^{x}+2 x \right ) x}+\frac {c_{1} x^{3}}{{\mathrm e}^{x}+2 x}+\frac {c_{2} x^{2}}{{\mathrm e}^{x}+2 x} \]
✓ Solution by Mathematica
Time used: 0.091 (sec). Leaf size: 43
DSolve[-x^(-5) + E^x*y[x] + 4*E^x*y'[x] + 6*E^x*y''[x] + 4*(2 + E^x)*Derivative[3][y][x] + (E^x + 2*x)*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1+24 x (x (x (c_4 x+c_3)+c_2)+c_1)}{24 x \left (2 x+e^x\right )} \\ \end{align*}