Internal problem ID [11058]
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: 233.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x -a \right )^{2} \left (x -b \right )^{2} y^{\prime \prime }+\left (x -a \right ) \left (x -b \right ) \left (2 x +\lambda \right ) y^{\prime }+\mu y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 143
dsolve((x-a)^2*(x-b)^2*diff(y(x),x$2)+(x-a)*(x-b)*(2*x+lambda)*diff(y(x),x)+mu*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (\left (\frac {a -x}{b -x}\right )^{-\frac {\sqrt {\lambda ^{2}+\left (2 a +2 b \right ) \lambda +a^{2}+2 a b +b^{2}-4 \mu }}{2 a -2 b}} c_{2} +\left (\frac {a -x}{b -x}\right )^{\frac {\sqrt {\lambda ^{2}+\left (2 a +2 b \right ) \lambda +a^{2}+2 a b +b^{2}-4 \mu }}{2 a -2 b}} c_{1} \right ) \left (\frac {b -x}{a -x}\right )^{\frac {a +b +\lambda }{2 a -2 b}} \]
✓ Solution by Mathematica
Time used: 2.299 (sec). Leaf size: 152
DSolve[(x-a)^2*(x-b)^2*y''[x]+(x-a)*(x-b)*(2*x+\[Lambda])*y'[x]+mu*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {(a+b+\lambda ) (\log (x-a)-\log (x-b))}{a-b}} \left (c_1 \exp \left (\frac {\left (\sqrt {\mu } \sqrt {\frac {(a+b+\lambda )^2}{\mu }-4}+a+b+\lambda \right ) (\log (x-a)-\log (x-b))}{2 (a-b)}\right )+c_2 \exp \left (\frac {\left (-\sqrt {\mu } \sqrt {\frac {(a+b+\lambda )^2}{\mu }-4}+a+b+\lambda \right ) (\log (x-a)-\log (x-b))}{2 (a-b)}\right )\right ) \]