Internal problem ID [11046]
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 A.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {\left (a^{2}+x^{2}\right )^{2} y^{\prime \prime }+y b^{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 83
dsolve((x^2+a^2)^2*diff(y(x),x$2)+b^2*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\left (\frac {i x -a}{i x +a}\right )^{\frac {\sqrt {a^{2}+b^{2}}}{2 a}} c_{1} +\left (\frac {i x -a}{i x +a}\right )^{-\frac {\sqrt {a^{2}+b^{2}}}{2 a}} c_{2} \right ) \sqrt {a^{2}+x^{2}} \]
✓ Solution by Mathematica
Time used: 0.489 (sec). Leaf size: 97
DSolve[(x^2+a^2)^2*y''[x]+b^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \sqrt {a^2+x^2} e^{-i \sqrt {\frac {b^2}{a^2}+1} \arctan \left (\frac {a}{x}\right )} \left (\frac {i c_2 e^{2 i \sqrt {\frac {b^2}{a^2}+1} \arctan \left (\frac {a}{x}\right )}}{a \sqrt {\frac {b^2}{a^2}+1}}+2 c_1\right ) \]