60.3.205 problem 1209

Internal problem ID [11215]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1209
Date solved : Monday, January 27, 2025 at 10:48:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 39

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

\[ y = \frac {\left (-c_{2} \pi \,\operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+c_{1} \right ) {\mathrm e}^{-\frac {x^{2}}{2}}+i \sqrt {2}\, \sqrt {\pi }\, c_{2} x}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 61

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

\[ y(x)\to \frac {-\sqrt {2 \pi } c_1 e^{-\frac {x^2}{2}} \text {erfi}\left (\frac {x}{\sqrt {2}}\right )+2 c_2 e^{-\frac {x^2}{2}-2}+2 c_1 x}{2 x^2} \]