Internal problem ID [11028]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 194.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 319
dsolve(x^2*(x+a__2)*diff(y(x),x$2)+x*(b__1*x+a__1)*diff(y(x),x)+(b__0*x+a__0)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{\frac {a_{2} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [\frac {a_{2} b_{1} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}, -\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -a_{2} b_{1} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}\right ], \left [\frac {a_{2} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )+c_{2} x^{-\frac {-a_{2} +a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [\frac {a_{2} b_{1} -a_{1} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}, -\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -a_{2} b_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}\right ], \left [\frac {a_{2} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right ) \]
✓ Solution by Mathematica
Time used: 1.236 (sec). Leaf size: 384
DSolve[x^2*(x+a2)*y''[x]+x*(b1*x+a1)*y'[x]+(b0*x+a0)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {a2}^{-\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}-\text {a1}+\text {a2}}{2 \text {a2}}} x^{-\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}+\text {a1}-\text {a2}}{2 \text {a2}}} \left (c_2 x^{\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}}{\text {a2}}} \operatorname {Hypergeometric2F1}\left (\frac {-\text {a1}+\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}-\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},\frac {-\text {a1}+\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}+\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2}^2-4 \text {a0} \text {a2}}}{\text {a2}},-\frac {x}{\text {a2}}\right )+c_1 \text {a2}^{\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}}{\text {a2}}} \operatorname {Hypergeometric2F1}\left (-\frac {\text {a1}-\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}+\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},-\frac {\text {a1}-\text {a2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {b0}}\right )+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}}{2 \text {a2}},1-\frac {\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2}^2-4 \text {a0} \text {a2}}}{\text {a2}},-\frac {x}{\text {a2}}\right )\right ) \]