Internal problem ID [11053]
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: 219.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (x^{2}+a \right ) y^{\prime \prime }+\left (b \,x^{2}+c \right ) x y^{\prime }+d y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 309
dsolve(x^2*(x^2+a)*diff(y(x),x$2)+(b*x^2+c)*x*diff(y(x),x)+d*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{\frac {a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [\frac {3 a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [\frac {2 a +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right )+c_{2} x^{-\frac {-a +c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {-3 a -c +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{4 a}, \frac {-\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}+\left (-2 b +5\right ) a +c}{4 a}\right ], \left [-\frac {-2 a +\sqrt {a^{2}+\left (-2 c -4 d \right ) a +c^{2}}}{2 a}\right ], -\frac {x^{2}}{a}\right ) \]
✓ Solution by Mathematica
Time used: 2.385 (sec). Leaf size: 336
DSolve[x^2*(x^2+a)*y''[x]+(b*x^2+c)*x*y'[x]+d*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to a^{-\frac {\sqrt {a^2-2 a (c+2 d)+c^2}+a-c}{4 a}} x^{-\frac {\sqrt {a^2-2 a (c+2 d)+c^2}-a+c}{2 a}} \left (c_2 x^{\frac {\sqrt {a^2-2 a (c+2 d)+c^2}}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {-2 b a+a+c-\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},\frac {a-c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},\frac {\sqrt {a^2-2 (c+2 d) a+c^2}}{2 a}+1,-\frac {x^2}{a}\right )+c_1 a^{\frac {\sqrt {a^2-2 a (c+2 d)+c^2}}{2 a}} \operatorname {Hypergeometric2F1}\left (-\frac {-a+c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},-\frac {-2 b a+a+c+\sqrt {a^2-2 (c+2 d) a+c^2}}{4 a},1-\frac {\sqrt {a^2-2 (c+2 d) a+c^2}}{2 a},-\frac {x^2}{a}\right )\right ) \]