Internal problem ID [11009]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 175.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (d -2 a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 1626
dsolve((a*x^2+b*x+c)*diff(y(x),x$2)+(d*x+k)*diff(y(x),x)+(d-2*a)*y(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 15.225 (sec). Leaf size: 164
DSolve[(a*x^2+b*x+c)*y''[x]+(d*x+k)*y'[x]+(d-2*a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (x (a x+b)+c)^{1-\frac {d}{2 a}} \exp \left (\frac {(b d-2 a k) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) \left (c_2 \int _1^x\exp \left (\frac {(d-4 a) \log (c+K[1] (b+a K[1]))-\frac {2 (b d-2 a k) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}}{2 a}\right )dK[1]+c_1\right ) \]