61.30.28 problem 176

Internal problem ID [12597]
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 : 176
Date solved : Wednesday, March 05, 2025 at 07:26:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k x +d \right ) y^{\prime }-k y&=0 \end{align*}

✓ Maple. Time used: 0.246 (sec). Leaf size: 315

ode:=(a*x^2+b*x+c)*diff(diff(y(x),x),x)+(k*x+d)*diff(y(x),x)-k*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_{1} \left (k x +d \right )+c_{2} {\left (2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, x \,a^{2}+\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, b a -4 a c +b^{2}\right )}^{\frac {a \left (a -\frac {k}{2}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+a d -\frac {b k}{2}}{\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}} \operatorname {hypergeom}\left (\left [-\frac {k \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a -2 a d +b k}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}, \frac {a \left (a +\frac {k}{2}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+a d -\frac {b k}{2}}{\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \left [\frac {4 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}-k \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a +2 a d -b k}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+4 a c -b^{2}}{8 a c -2 b^{2}}\right ) \]

✓ Mathematica. Time used: 1.784 (sec). Leaf size: 90

ode=(a*x^2+b*x+c)*D[y[x],{x,2}]+(k*x+d)*D[y[x],x]-k*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {(d+k x) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {d^2+2 k K[1] d+2 c k+k K[1] (2 b+(2 a+k) K[1])}{(d+k K[1]) (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right )}{d} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-k*y(x) + (d + k*x)*Derivative(y(x), x) + (a*x**2 + b*x + c)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False