Internal problem ID [11063]
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: 228.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (2 a x +c \right ) \left (a \,x^{2}+b \right ) y^{\prime }+k y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 162
dsolve((a*x^2+b)^2*diff(y(x),x$2)+(2*a*x+c)*(a*x^2+b)*diff(y(x),x)+k*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\frac {\sqrt {a b}+i a x}{i a x -\sqrt {a b}}\right )^{\frac {i \sqrt {a b}\, c \sqrt {-a b}+a^{2} \sqrt {\frac {c^{2}-4 k}{a^{2}}}\, b}{4 a b \sqrt {-a b}}}+c_{2} \left (\frac {\sqrt {a b}+i a x}{i a x -\sqrt {a b}}\right )^{\frac {i \sqrt {a b}\, c \sqrt {-a b}-a^{2} \sqrt {\frac {c^{2}-4 k}{a^{2}}}\, b}{4 a b \sqrt {-a b}}} \]
✓ Solution by Mathematica
Time used: 2.188 (sec). Leaf size: 91
DSolve[(a*x^2+b)^2*y''[x]+(2*a*x+c)*(a*x^2+b)*y'[x]+k*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\left (\sqrt {c^2-4 k}+c\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}} \left (c_2 e^{\frac {\sqrt {c^2-4 k} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}}+c_1\right ) \]