Internal problem ID [11067]
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: 232.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x -a \right )^{2} \left (x -b \right )^{2} y^{\prime \prime }-c y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 116
dsolve((x-a)^2*(x-b)^2*diff(y(x),x$2)-c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {\left (-x +a \right ) \left (-x +b \right )}\, \left (\frac {-x +a}{-x +b}\right )^{\frac {\sqrt {a^{2}-2 a b +b^{2}+4 c}}{2 a -2 b}}+c_{2} \sqrt {\left (-x +a \right ) \left (-x +b \right )}\, \left (\frac {-x +a}{-x +b}\right )^{-\frac {\sqrt {a^{2}-2 a b +b^{2}+4 c}}{2 a -2 b}} \]
✓ Solution by Mathematica
Time used: 1.085 (sec). Leaf size: 141
DSolve[(x-a)^2*(x-b)^2*y''[x]-c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (x-a)^{\frac {1}{2} \left (1-\sqrt {\frac {4 c}{(a-b)^2}+1}\right )} (x-b)^{\frac {1}{2} \left (1-\sqrt {\frac {4 c}{(a-b)^2}+1}\right )} \left (c_1 (x-a)^{\sqrt {\frac {4 c}{(a-b)^2}+1}}-\frac {c_2 (x-b)^{\sqrt {\frac {4 c}{(a-b)^2}+1}}}{(a-b) \sqrt {\frac {4 c}{(a-b)^2}+1}}\right ) \]