60.3.207 problem 1211

Internal problem ID [11217]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1211
Date solved : Monday, January 27, 2025 at 10:49:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+4 x^{3} y^{\prime }+\left (4 x^{4}+2 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 33

dsolve(x^2*diff(diff(y(x),x),x)+4*x^3*diff(y(x),x)+(4*x^4+2*x^2+1)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.074 (sec). Leaf size: 60

DSolve[(1 + 2*x^2 + 4*x^4)*y[x] + 4*x^3*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} e^{-x^2} x^{\frac {1}{2}-\frac {i \sqrt {3}}{2}} \left (3 c_1-i \sqrt {3} c_2 x^{i \sqrt {3}}\right ) \]