61.32.7 problem 217
Internal
problem
ID
[12717]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-7
Problem
number
:
217
Date
solved
:
Tuesday, January 28, 2025 at 04:16:30 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} \left (x -a \right )^{2} y^{\prime \prime }+b y&=c \,x^{2} \left (x -a \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.020 (sec). Leaf size: 219
dsolve(x^2*(x-a)^2*diff(y(x),x$2)+b*y(x)=c*x^2*(x-a)^2,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: 0.182 (sec). Leaf size: 280
DSolve[x^2*(x-a)^2*D[y[x],{x,2}]+b*y[x]==c*x^2*(x-a)^2,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \exp \left (\int _1^x\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \left (\int _1^x-c \exp \left (\int _1^{K[3]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right ) \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2]dK[3]+\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[2] \left (\int _1^xc \exp \left (\int _1^{K[4]}\frac {\sqrt {1-\frac {4 b}{a^2}} a+a-2 K[1]}{2 (a-K[1]) K[1]}dK[1]\right )dK[4]+c_2\right )+c_1\right )
\]