Internal problem ID [10945]
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: 121.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 177
dsolve(x^2*diff(y(x),x$2)+(a*x^(3*n)+b*x^(2*n)+1/4-1/4*n^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\frac {2 \,3^{\frac {5}{6}} \pi c_{2} \left (a \,x^{n}+b \right ) \operatorname {BesselI}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {-x^{3 n} a^{3}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right )}{3}+c_{1} \operatorname {BesselI}\left (-\frac {1}{3}, \frac {2 \sqrt {\frac {-x^{3 n} a^{3}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}} {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{a^{2} n^{2}}\right )}^{\frac {1}{3}}\right ) x^{-\frac {n}{2}+\frac {1}{2}}}{3 {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{a^{2} n^{2}}\right )}^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x^2*y''[x]+(a*x^(3*n)+b*x^(2*n)+1/4-1/4*n^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved