61.30.33 problem 181

Internal problem ID [12602]
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 : 181
Date solved : Thursday, March 13, 2025 at 11:52:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (k^{3}+x^{3}\right ) y^{\prime }-\left (k^{2}-k x +x^{2}\right ) y&=0 \end{align*}

✓ Maple. Time used: 1.379 (sec). Leaf size: 246

ode:=(a*x^2+b*x+c)*diff(diff(y(x),x),x)+(k^3+x^3)*diff(y(x),x)-(k^2-k*x+x^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x +k \right ) \left (\left (\int \frac {\left (2 a x +b -\sqrt {-4 a c +b^{2}}\right )^{-\frac {k^{3}}{\sqrt {-4 a c +b^{2}}}} {\left (\frac {-2 a x +\sqrt {-4 a c +b^{2}}-b}{2 a x +b +\sqrt {-4 a c +b^{2}}}\right )}^{-\frac {3 b c}{2 a^{2} \sqrt {-4 a c +b^{2}}}} {\left (\frac {2 a x +b +\sqrt {-4 a c +b^{2}}}{-2 a x +\sqrt {-4 a c +b^{2}}-b}\right )}^{-\frac {b^{3}}{2 a^{3} \sqrt {-4 a c +b^{2}}}} \left (a \,x^{2}+b x +c \right )^{\frac {a c -b^{2}}{2 a^{3}}} {\left (\frac {2 a x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}\right )}^{\frac {k^{3}}{\sqrt {-4 a c +b^{2}}}} {\mathrm e}^{-\frac {x \left (a x -2 b \right )}{2 a^{2}}}}{\left (x +k \right )^{2}}d x \right ) c_{2} +c_{1} \right ) \]

✓ Mathematica. Time used: 1.064 (sec). Leaf size: 91

ode=(a*x^2+b*x+c)*D[y[x],{x,2}]+(x^3+k^3)*D[y[x],x]-(x^2-k*x+k^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {(k+x) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {k^4+K[1] k^3+K[1]^3 k+K[1]^4+2 a K[1]^2+2 c+2 b K[1]}{(k+K[1]) (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right )}{k} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
y = Function("y") 
ode = Eq((k**3 + x**3)*Derivative(y(x), x) - (k**2 - k*x + x**2)*y(x) + (a*x**2 + b*x + c)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False