61.30.31 problem 179

Internal problem ID [12600]
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
Problem number : 179
Date solved : Monday, March 31, 2025 at 05:53:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a_{2} x^{2}+b_{2} x +c_{2} \right ) y^{\prime \prime }+\left (b_{1} x +c_{1} \right ) y^{\prime }+c_{0} y&=0 \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 501

ode:=(a__2*x^2+b__2*x+c__2)*diff(diff(y(x),x),x)+(b__1*x+c__1)*diff(y(x),x)+c__0*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_3 \operatorname {hypergeom}\left (\left [-\frac {a_{2} -b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}, \frac {-a_{2} +b_{1} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}}{2 a_{2}}\right ], \left [\frac {b_{1} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} -2 a_{2} c_{1} +b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 a_{2} c_{2} -b_{2}^{2}}{8 a_{2} c_{2} -2 b_{2}^{2}}\right )+c_4 {\left (2 \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, x \,a_{2}^{2}+\sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, b_{2} a_{2} -4 a_{2} c_{2} +b_{2}^{2}\right )}^{\frac {a_{2} \left (a_{2} -\frac {b_{1}}{2}\right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+a_{2} c_{1} -\frac {b_{1} b_{2}}{2}}{\sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}} \operatorname {hypergeom}\left (\left [\frac {a_{2} \left (a_{2} -\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+2 a_{2} c_{1} -b_{1} b_{2}}{2 \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}, \frac {a_{2} \left (a_{2} +\sqrt {a_{2}^{2}+\left (-2 b_{1} -4 c_{0} \right ) a_{2} +b_{1}^{2}}\right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+2 a_{2} c_{1} -b_{1} b_{2}}{2 \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2}^{2}}\right ], \left [\frac {4 a_{2}^{2} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}-b_{1} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}\, a_{2} +2 a_{2} c_{1} -b_{1} b_{2}}{2 a_{2}^{2} \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}}\right ], \frac {\left (-2 a_{2}^{2} x -a_{2} b_{2} \right ) \sqrt {\frac {-4 a_{2} c_{2} +b_{2}^{2}}{a_{2}^{2}}}+4 a_{2} c_{2} -b_{2}^{2}}{8 a_{2} c_{2} -2 b_{2}^{2}}\right ) \]

✓ Mathematica. Time used: 4.762 (sec). Leaf size: 498

ode=(a2*x^2+b2*x+c2)*D[y[x],{x,2}]+(b1*x+c1)*D[y[x],x]+c0*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {\text {a2}-\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {-\text {a2}+\text {b1}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}}{2 \text {a2}},\frac {\text {b1} \left (\text {b2}+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}\right )-2 \text {a2} \text {c1}}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )-c_2 2^{\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}-\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-1} \exp \left (-\frac {i \pi \left (\text {b1} \left (\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+\text {b2}\right )-2 \text {a2} \text {c1}\right )}{2 \text {a2} \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \left (\frac {\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}+2 \text {a2} x+\text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right )^{-\frac {\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}}{2 \text {a2}}+\frac {\text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+1} \operatorname {Hypergeometric2F1}\left (\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}-\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},\frac {\frac {2 \text {c1} \text {a2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {a2}+\sqrt {(\text {a2}-\text {b1})^2-4 \text {a2} \text {c0}}-\frac {\text {b1} \text {b2}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}}{2 \text {a2}},-\frac {\frac {\text {b2} \text {b1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}+\text {b1}+\text {a2} \left (-\frac {2 \text {c1}}{\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}-4\right )}{2 \text {a2}},\frac {\text {b2}+2 \text {a2} x+\sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}{2 \sqrt {\text {b2}^2-4 \text {a2} \text {c2}}}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a__2 = symbols("a__2") 
b__1 = symbols("b__1") 
b__2 = symbols("b__2") 
c__0 = symbols("c__0") 
c__1 = symbols("c__1") 
c__2 = symbols("c__2") 
y = Function("y") 
ode = Eq(c__0*y(x) + (b__1*x + c__1)*Derivative(y(x), x) + (a__2*x**2 + b__2*x + c__2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False