54.3.241 problem 1257
Internal
problem
ID
[12536]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1257
Date
solved
:
Wednesday, October 01, 2025 at 01:57:17 AM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} x \left (x -1\right ) y^{\prime \prime }+\left (\left (a +1\right ) x +b \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 27
ode:=x*(x-1)*diff(diff(y(x),x),x)+((a+1)*x+b)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 +x^{b +1} \operatorname {hypergeom}\left (\left [b +1, b +a +1\right ], \left [2+b \right ], x\right ) c_2
\]
✓ Mathematica. Time used: 0.034 (sec). Leaf size: 48
ode=(b + (1 + a)*x)*D[y[x],x] + (-1 + x)*x*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \int _1^x\exp \left (\int _1^{K[2]}\frac {b+a K[1]+K[1]}{K[1]-K[1]^2}dK[1]\right ) c_1dK[2]+c_2 \end{align*}
✓ Sympy. Time used: 0.305 (sec). Leaf size: 139
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(x*(x - 1)*Derivative(y(x), (x, 2)) + (b + x*(a + 1))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + x^{\frac {- \left (\operatorname {re}{\left (x\right )} - 1\right ) \left (\operatorname {re}{\left (b\right )} + \operatorname {re}{\left (a x\right )} + 1\right ) - \left (\operatorname {im}{\left (b\right )} + \operatorname {im}{\left (a x\right )}\right ) \operatorname {im}{\left (x\right )}}{\left (\operatorname {re}{\left (x\right )} - 1\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2}}} \left (C_{2} \sin {\left (\frac {\log {\left (x \right )} \left |{\left (\operatorname {re}{\left (x\right )} - 1\right ) \left (\operatorname {im}{\left (b\right )} + \operatorname {im}{\left (a x\right )}\right ) - \left (\operatorname {re}{\left (b\right )} + \operatorname {re}{\left (a x\right )} + 1\right ) \operatorname {im}{\left (x\right )}}\right |}{\left (\operatorname {re}{\left (x\right )} - 1\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2}} \right )} + C_{3} \cos {\left (\frac {\left (\left (\operatorname {re}{\left (x\right )} - 1\right ) \left (\operatorname {im}{\left (b\right )} + \operatorname {im}{\left (a x\right )}\right ) - \left (\operatorname {re}{\left (b\right )} + \operatorname {re}{\left (a x\right )} + 1\right ) \operatorname {im}{\left (x\right )}\right ) \log {\left (x \right )}}{\left (\operatorname {re}{\left (x\right )} - 1\right )^{2} + \left (\operatorname {im}{\left (x\right )}\right )^{2}} \right )}\right )
\]