54.3.318 problem 1335

Internal problem ID [12613]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1335
Date solved : Friday, October 03, 2025 at 03:43:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (3 x -1\right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {\left (a x +b \right ) y}{4 x \left (x -1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 57

ode:=diff(diff(y(x),x),x) = -1/2/x*(3*x-1)/(x-1)*diff(y(x),x)-1/4*(a*x+b)/x/(x-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {LegendreP}\left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-b -a}, \sqrt {x}\right )+c_2 \operatorname {LegendreQ}\left (\frac {\sqrt {1-4 a}}{2}-\frac {1}{2}, \sqrt {-b -a}, \sqrt {x}\right ) \]

✓ Mathematica. Time used: 1.201 (sec). Leaf size: 529

ode=D[y[x],{x,2}] == -1/4*((b + a*x)*y[x])/((-1 + x)^2*x) - ((-1 + 3*x)*D[y[x],x])/(2*(-1 + x)*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt [4]{x} (x-1)^{\frac {2 a \sqrt {-4 \sqrt {(4 a-1) (a+b)}-8 a-4 b+1}+2 b \left (\sqrt {-4 \sqrt {(4 a-1) (a+b)}-8 a-4 b+1}+2\right )-\sqrt {(4 a-1) (a+b)} \sqrt {-4 \sqrt {(4 a-1) (a+b)}-8 a-4 b+1}+1}{8 b+2}} e^{-\frac {1}{4} \int \frac {1-3 x}{x-x^2} \, dx} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+1\right ),\frac {8 \sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1} a+4 b \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+1\right )-4 \sqrt {(4 a-1) (a+b)} \sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}-\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+1}{16 b+4},\frac {1}{2},x\right )+i c_2 \sqrt {x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+3\right ),\frac {8 \sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1} a+4 b \left (\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+3\right )-4 \sqrt {(4 a-1) (a+b)} \sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}-\sqrt {-8 a-4 b-4 \sqrt {(4 a-1) (a+b)}+1}+3}{16 b+4},\frac {3}{2},x\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (3*x - 1)*Derivative(y(x), x)/(2*x*(x - 1)) + (a*x + b)*y(x)/(4*x*(x - 1)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False