60.3.366 problem 1383

Internal problem ID [11362]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1383
Date solved : Sunday, March 30, 2025 at 08:18:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

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

\[ y = c_1 \left (\frac {-x +a}{-x +b}\right )^{\beta }+c_2 \left (\frac {-x +a}{-x +b}\right )^{\alpha } \]

✓ Mathematica. Time used: 2.903 (sec). Leaf size: 187

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

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

✗ Sympy

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

False