60.3.55 problem 1056

Internal problem ID [11065]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1056
Date solved : Monday, January 27, 2025 at 10:43:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }+y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 56

dsolve(diff(diff(y(x),x),x)-x^2*diff(y(x),x)+x*y(x)=0,y(x), singsol=all)
 

\[ y = -\frac {\left (-c_{2} 3^{{1}/{3}} {\mathrm e}^{\frac {x^{3}}{3}}-c_{1} x \right ) \left (-x^{3}\right )^{{2}/{3}}+c_{2} x^{3} \left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3}, -\frac {x^{3}}{3}\right )\right )}{\left (-x^{3}\right )^{{2}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.070 (sec). Leaf size: 41

DSolve[x*y[x] - x^2*D[y[x],x] + D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 x-\frac {c_2 \sqrt [3]{-x^3} \Gamma \left (-\frac {1}{3},-\frac {x^3}{3}\right )}{3 \sqrt [3]{3}} \]