Internal problem ID [11041]
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: 217.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} \left (x -a \right )^{2} y^{\prime \prime }+y b=c \,x^{2} \left (x -a \right )^{2}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 219
dsolve(x^2*(x-a)^2*diff(y(x),x$2)+b*y(x)=c*x^2*(x-a)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {x \left (a -x \right )}\, \left (\left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{1} \sqrt {a^{2}-4 b}+\left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{2} \sqrt {a^{2}-4 b}+\left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \left (\int \sqrt {x \left (a -x \right )}\, \left (\frac {x}{a -x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) c -\left (\int \sqrt {x \left (a -x \right )}\, \left (\frac {a -x}{x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) \left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c \right )}{\sqrt {a^{2}-4 b}} \]
✓ Solution by Mathematica
Time used: 0.958 (sec). Leaf size: 371
DSolve[x^2*(x-a)^2*y''[x]+b*y[x]==c*x^2*(x-a)^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {a c x^2 (a-x) \left (1-\frac {x}{a}\right )^{-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2}} \left (\left (\sqrt {1-\frac {4 b}{a^2}}-3\right ) \left (1-\frac {x}{a}\right )^{\sqrt {1-\frac {4 b}{a^2}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {3}{2},\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {5}{2},\frac {x}{a}\right )+\left (\sqrt {1-\frac {4 b}{a^2}}+3\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}-\frac {1}{2},\frac {3}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}},\frac {5}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}},\frac {x}{a}\right )\right )}{2 \left (2 a^2+b\right ) \sqrt {1-\frac {4 b}{a^2}}}+c_1 x^{\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {1}{2}} (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}}+\frac {c_2 x^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}} (x-a)^{\frac {1}{2} \sqrt {1-\frac {4 b}{a^2}}+\frac {1}{2}}}{a \sqrt {1-\frac {4 b}{a^2}}} \]