Internal problem ID [10950]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 116.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
dsolve(x^2*diff(y(x),x$2)-(a*x^3+5/16)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \sinh \left (\frac {2 x^{\frac {3}{2}} \sqrt {a}}{3}\right )}{x^{\frac {1}{4}}}+\frac {c_{2} \cosh \left (\frac {2 x^{\frac {3}{2}} \sqrt {a}}{3}\right )}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.102 (sec). Leaf size: 60
DSolve[x^2*y''[x]-(a*x^3+5/16)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\frac {2}{3} \sqrt {a} x^{3/2}} \left (2 c_1 e^{\frac {4}{3} \sqrt {a} x^{3/2}}-\frac {c_2}{\sqrt {a}}\right )}{2 \sqrt [4]{x}} \]