60.3.375 problem 1381
Internal
problem
ID
[11385]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1381
Date
solved
:
Monday, January 27, 2025 at 11:18:49 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*}
✓ Solution by Maple
Time used: 0.020 (sec). Leaf size: 219
dsolve(diff(diff(y(x),x),x) = -b/x^2/(x-a)^2*y(x)+c,y(x), singsol=all)
\[
y = \frac {\left (-\left (\int \sqrt {x \left (a -x \right )}\, \left (\frac {a -x}{x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) \left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c +\left (\int \sqrt {x \left (a -x \right )}\, \left (\frac {x}{a -x}\right )^{-\frac {\sqrt {a^{2}-4 b}}{2 a}}d x \right ) \left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c +\left (\frac {x}{a -x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{1} \sqrt {a^{2}-4 b}+\left (\frac {a -x}{x}\right )^{\frac {\sqrt {a^{2}-4 b}}{2 a}} c_{2} \sqrt {a^{2}-4 b}\right ) \sqrt {x \left (a -x \right )}}{\sqrt {a^{2}-4 b}}
\]
✓ Solution by Mathematica
Time used: 0.189 (sec). Leaf size: 280
DSolve[D[y[x],{x,2}] == c - (b*y[x])/(x^2*(-a + x)^2),y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \exp \left (\int _1^x\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \left (\int _1^x-c \exp \left (\int _1^{K[3]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2]dK[3]+\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2] \left (\int _1^xc \exp \left (\int _1^{K[4]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[4]+c_2\right )+c_1\right )
\]