Internal problem ID [11057]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 222 B.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-a^{2}+x^{2}\right )^{2} y^{\prime \prime }+y b^{2}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 87
dsolve((x^2-a^2)^2*diff(y(x),x$2)+b^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {\left (-x +a \right ) \left (x +a \right )}\, \left (\frac {-x +a}{x +a}\right )^{\frac {\sqrt {a^{2}-b^{2}}}{2 a}}+c_{2} \sqrt {\left (-x +a \right ) \left (x +a \right )}\, \left (\frac {-x +a}{x +a}\right )^{-\frac {\sqrt {a^{2}-b^{2}}}{2 a}} \]
✓ Solution by Mathematica
Time used: 0.529 (sec). Leaf size: 142
DSolve[(x^2-a^2)^2*y''[x]+b^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}} (a+x)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}} \left (2 a c_1 \sqrt {1-\frac {b^2}{a^2}} (x-a)^{\sqrt {1-\frac {b^2}{a^2}}}-c_2 (a+x)^{\sqrt {1-\frac {b^2}{a^2}}}\right )}{2 a \sqrt {1-\frac {b^2}{a^2}}} \]