54.3.324 problem 1341

Internal problem ID [12619]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1341
Date solved : Friday, October 03, 2025 at 03:43:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 194

ode:=diff(diff(y(x),x),x) = -1/x*(2*a*x+b)/(a*x+b)*diff(y(x),x)-(a*v*x-b)/(a*x+b)/x^2*y(x)+A*x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{-\frac {\sqrt {1-4 v}}{2}} a^{2} c_1 \left (v +6\right ) \left (v +2\right ) \left (v +12\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}+\frac {\sqrt {1-4 v}}{2}\right ], \left [1+\sqrt {1-4 v}\right ], -\frac {b}{a x}\right )-3 A \,b^{2} \left (v +4\right ) x^{{3}/{2}}+\left (v +2\right ) \left (A b \left (v +4\right ) x^{{5}/{2}}+\left (A \,x^{{7}/{2}}+\operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}-\frac {\sqrt {1-4 v}}{2}\right ], \left [1-\sqrt {1-4 v}\right ], -\frac {b}{a x}\right ) \left (v +12\right ) c_2 \,x^{\frac {\sqrt {1-4 v}}{2}}\right ) \left (v +6\right ) a \right ) a}{\sqrt {x}\, a^{2} \left (v +6\right ) \left (v +2\right ) \left (v +12\right )} \]

✓ Mathematica. Time used: 21.607 (sec). Leaf size: 725

ode=D[y[x],{x,2}] == A*x - ((-b + a*v*x)*y[x])/(x^2*(b + a*x)) - ((b + 2*a*x)*D[y[x],x])/(x*(b + a*x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (A*a*x**4 + A*b*x**3 - a*v*x*y(x) - a*x**3*Derivative(y(x), (x, 2)) - b*x**2*Derivative(y(x), (x, 2)) + b*y(x))/(x*(2*a*x + b)) cannot be solved by the factorable group method