60.3.403 problem 1409

Internal problem ID [11413]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1409
Date solved : Monday, January 27, 2025 at 11:20:15 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }&=-a \,x^{2 a -1} x^{-2 a} y^{\prime }-b^{2} x^{-2 a} y \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 39

dsolve(diff(diff(y(x),x),x) = -a*x^(2*a-1)/(x^(2*a))*diff(y(x),x)-b^2/(x^(2*a))*y(x),y(x), singsol=all)
 

\[ y = c_{1} \sin \left (\frac {b \,x^{-a +1}}{a -1}\right )+c_{2} \cos \left (\frac {b \,x^{-a +1}}{a -1}\right ) \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \cos \left (\frac {b x^{1-a}}{a-1}\right )+c_2 \sin \left (\frac {b x^{1-a}}{1-a}\right ) \]