60.3.440 problem 1447

Internal problem ID [11450]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1447
Date solved : Monday, January 27, 2025 at 11:22:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {y^{\prime }}{x}-\frac {\left (-1-x \right ) y}{x^{4}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 20

dsolve(diff(diff(y(x),x),x) = -1/x*diff(y(x),x)-(-x-1)/x^4*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.088 (sec). Leaf size: 29

DSolve[D[y[x],{x,2}] == -(((-1 - x)*y[x])/x^4) - D[y[x],x]/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\frac {1}{x}-\frac {1}{2}} \left (c_1-e c_2 \operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right )\right ) \]