60.3.364 problem 1381
Internal
problem
ID
[11360]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1381
Date
solved
:
Sunday, March 30, 2025 at 08:18:15 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }&=-\frac {b y}{x^{2} \left (x -a \right )^{2}}+c \end{align*}
✓ Maple. Time used: 0.105 (sec). Leaf size: 219
ode:=diff(diff(y(x),x),x) = -b/x^2/(x-a)^2*y(x)+c;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\sqrt {x \left (-x +a \right )}\, \left (\left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_2 \sqrt {a^{2}-4 b}-\left (\frac {-x +a}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \int \sqrt {x \left (-x +a \right )}\, \left (\frac {-x +a}{x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x c +\left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_1 \sqrt {a^{2}-4 b}+\left (\frac {x}{-x +a}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} \int \sqrt {x \left (-x +a \right )}\, \left (\frac {x}{-x +a}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x c \right )}{\sqrt {a^{2}-4 b}}
\]
✓ Mathematica. Time used: 0.189 (sec). Leaf size: 280
ode=D[y[x],{x,2}] == c - (b*y[x])/(x^2*(-a + x)^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \exp \left (\int _1^x\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \left (\int _1^x-c \exp \left (\int _1^{K[3]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2]dK[3]+\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2] \left (\int _1^xc \exp \left (\int _1^{K[4]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[4]+c_2\right )+c_1\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(b*y(x)/(x**2*(-a + x)**2) - c + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve b*y(x)/(x**2*(-a + x)**2) - c + Derivative(y(x), (x, 2))