Internal problem ID [11059]
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: 224.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+2 a x \left (a \,x^{2}+b \right ) y^{\prime }+c y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 47
dsolve((a*x^2+b)^2*diff(y(x),x$2)+2*a*x*(a*x^2+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sin \left (\frac {\sqrt {c}\, \arctan \left (\frac {x a}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )+c_{2} \cos \left (\frac {\sqrt {c}\, \arctan \left (\frac {x a}{\sqrt {a b}}\right )}{\sqrt {a b}}\right ) \]
✓ Solution by Mathematica
Time used: 2.133 (sec). Leaf size: 72
DSolve[(a*x^2+b)^2*y''[x]+2*a*x*(a*x^2+b)*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cos \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right )+c_2 \sin \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right ) \]