60.3.204 problem 1208

Internal problem ID [11214]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1208
Date solved : Monday, January 27, 2025 at 10:48:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{3} y^{\prime }+\left (x^{2}-2\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 35

dsolve(x^2*diff(diff(y(x),x),x)+x^3*diff(y(x),x)+(x^2-2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.110 (sec). Leaf size: 49

DSolve[(-2 + x^2)*y[x] + x^3*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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