60.7.84 problem 1675 (book 6.84)

Internal problem ID [11673]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1675 (book 6.84)
Date solved : Monday, January 27, 2025 at 11:29:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 a +b -1\right ) x y^{\prime }+\left (c^{2} b^{2} x^{2 b}+a \left (a +b \right )\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.107 (sec). Leaf size: 25

dsolve(x^2*diff(diff(y(x),x),x)-(2*a+b-1)*x*diff(y(x),x)+(c^2*b^2*x^(2*b)+a*(a+b))*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.089 (sec). Leaf size: 69

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

\[ y(x)\to 2^{-\frac {a+b}{b}} c^{a/b} \left (x^{2 b}\right )^{\frac {a}{2 b}} \left (2 c_1 \cos \left (c \sqrt {x^{2 b}}\right )+c_2 \sin \left (c \sqrt {x^{2 b}}\right )\right ) \]