60.3.370 problem 1376

Internal problem ID [11380]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1376
Date solved : Monday, January 27, 2025 at 11:18:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 81

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

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

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 69

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

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