61.32.24 problem 233
Internal
problem
ID
[12655]
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
:
233
Date
solved
:
Monday, March 31, 2025 at 06:49:35 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (x -a \right )^{2} \left (x -b \right )^{2} y^{\prime \prime }+\left (x -a \right ) \left (x -b \right ) \left (2 x +\lambda \right ) y^{\prime }+\mu y&=0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 143
ode:=(x-a)^2*(x-b)^2*diff(diff(y(x),x),x)+(x-a)*(x-b)*(2*x+lambda)*diff(y(x),x)+mu*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (\left (\frac {-x +a}{-x +b}\right )^{-\frac {\sqrt {\lambda ^{2}+\left (2 a +2 b \right ) \lambda +a^{2}+2 a b +b^{2}-4 \mu }}{2 a -2 b}} c_2 +\left (\frac {-x +a}{-x +b}\right )^{\frac {\sqrt {\lambda ^{2}+\left (2 a +2 b \right ) \lambda +a^{2}+2 a b +b^{2}-4 \mu }}{2 a -2 b}} c_1 \right ) \left (\frac {-x +b}{-x +a}\right )^{\frac {b +a +\lambda }{2 a -2 b}}
\]
✓ Mathematica. Time used: 31.506 (sec). Leaf size: 565
ode=(x-a)^2*(x-b)^2*D[y[x],{x,2}]+(x-a)*(x-b)*(2*x+\[Lambda])*D[y[x],x]+mu*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \left (\frac {b (a-x)}{a (b-x)}\right )^{-\frac {a b \left (2 a \sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}-2 b \sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}-3 a-3 b\right )}{(a-b)^3}} \left (\frac {a (b-x)}{b (a-x)}\right )^{-\frac {(a+b)^2 \left (a \left (\sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}-1\right )-b \left (\sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}+1\right )\right )}{2 (a-b)^3}} \exp \left (-\frac {(b-x) \left (\frac {x^4 \operatorname {AppellF1}\left (4,2,2,5,\frac {x}{a},\frac {x}{b}\right )}{a^2 b^2}-2 \int _1^x\frac {\lambda +2 K[1]}{(K[1]-a) (K[1]-b)}dK[1]\right )+\frac {4 x \left (x \left (a^2+b^2\right )-a b (a+b)\right )}{(a-b)^2 (a-x)}}{4 (x-b)}\right ) \left (c_2 \int _1^x\exp \left (\frac {2 K[2] \left ((b-K[2]) a^2+b^2 a-b^2 K[2]\right )}{(a-b)^2 (a-K[2]) (b-K[2])}-\frac {\operatorname {AppellF1}\left (4,2,2,5,\frac {K[2]}{a},\frac {K[2]}{b}\right ) K[2]^4}{2 a^2 b^2}\right ) \left (\frac {b (a-K[2])}{a (b-K[2])}\right )^{\frac {2 a b \left (2 \sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}} a-3 a-3 b-2 b \sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}\right )}{(a-b)^3}} \left (\frac {a (b-K[2])}{b (a-K[2])}\right )^{\frac {(a+b)^2 \left (a \left (\sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}-1\right )-b \left (\sqrt {\frac {(a+b+\lambda )^2-4 \mu }{(a-b)^2}}+1\right )\right )}{(a-b)^3}}dK[2]+c_1\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
lambda_ = symbols("lambda_")
mu = symbols("mu")
y = Function("y")
ode = Eq(mu*y(x) + (-a + x)**2*(-b + x)**2*Derivative(y(x), (x, 2)) + (-a + x)*(-b + x)*(lambda_ + 2*x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
False