Internal problem ID [11072]
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: 237.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{2}+b x +c \right )^{2} y^{\prime \prime }+\left (2 a x +k \right ) \left (a \,x^{2}+b x +c \right ) y^{\prime }+y m=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 375
dsolve((a*x^2+b*x+c)^2*diff(y(x),x$2)+(2*a*x+k)*(a*x^2+b*x+c)*diff(y(x),x)+m*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\left (\frac {2 a x +\sqrt {-4 a c +b^{2}}+b}{-2 a x +\sqrt {-4 a c +b^{2}}-b}\right )}^{-\frac {b}{2 \sqrt {-4 a c +b^{2}}}} {\left (\frac {-2 a x +\sqrt {-4 a c +b^{2}}-b}{2 a x +\sqrt {-4 a c +b^{2}}+b}\right )}^{-\frac {k}{2 \sqrt {-4 a c +b^{2}}}} {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {\frac {b^{2}-2 b k +k^{2}-4 m}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+c_{2} {\left (\frac {2 a x +\sqrt {-4 a c +b^{2}}+b}{-2 a x +\sqrt {-4 a c +b^{2}}-b}\right )}^{-\frac {b}{2 \sqrt {-4 a c +b^{2}}}} {\left (\frac {-2 a x +\sqrt {-4 a c +b^{2}}-b}{2 a x +\sqrt {-4 a c +b^{2}}+b}\right )}^{-\frac {k}{2 \sqrt {-4 a c +b^{2}}}} {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {\frac {b^{2}-2 b k +k^{2}-4 m}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} \]
✓ Solution by Mathematica
Time used: 2.382 (sec). Leaf size: 157
DSolve[(a*x^2+b*x+c)^2*y''[x]+(2*a*x+k)*(a*x^2+b*x+c)*y'[x]+m*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \exp \left (\frac {\left (-\sqrt {m} \sqrt {\frac {b^2-2 b k+k^2-4 m}{m}}+b-k\right ) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right )+c_2 \exp \left (\frac {\left (\sqrt {m} \sqrt {\frac {b^2-2 b k+k^2-4 m}{m}}+b-k\right ) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}\right ) \]