29.31.31 problem 932
Internal
problem
ID
[5510]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
31
Problem
number
:
932
Date
solved
:
Sunday, March 30, 2025 at 08:28:55 AM
CAS
classification
:
[_quadrature]
\begin{align*} 4 x \left (a -x \right ) \left (b -x \right ) {y^{\prime }}^{2}&=\left (a b -2 x \left (a +b \right )+2 x^{2}\right )^{2} \end{align*}
✓ Maple. Time used: 0.049 (sec). Leaf size: 85
ode:=4*x*(a-x)*(b-x)*diff(y(x),x)^2 = (a*b-2*x*(a+b)+2*x^2)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\int \frac {2 x^{2}+\left (-2 a -2 b \right ) x +a b}{\sqrt {x \left (b -x \right ) \left (a -x \right )}}d x}{2}+c_1 \\
y &= \frac {\int \frac {2 x^{2}+\left (-2 a -2 b \right ) x +a b}{\sqrt {x \left (b -x \right ) \left (a -x \right )}}d x}{2}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 15.724 (sec). Leaf size: 375
ode=4 x(a-x)(b-x) (D[y[x],x])^2==(a b-2 x(a+b)+2 x^2)^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to c_1-\frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}} \\
y(x)\to \frac {(a-x) \left (2 \left (a^2-b^2\right ) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} E\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )+b (a+2 b) \sqrt {\frac {x}{a}} \sqrt {\frac {x-b}{a-b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {x}{a}-1}\right ),\frac {a}{a-b}\right )+2 i x \sqrt {\frac {x}{a}-1} (b-x)\right )}{3 \sqrt {\frac {x}{a}-1} \sqrt {x (a-x) (x-b)}}+c_1 \\
\end{align*}
✓ Sympy. Time used: 22.146 (sec). Leaf size: 214
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(4*x*(a - x)*(b - x)*Derivative(y(x), x)**2 - (a*b + 2*x**2 - 2*x*(a + b))**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {a b \int \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx}{2} + a \int x \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx + b \int x \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx - \int x^{2} \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx, \ y{\left (x \right )} = C_{1} + \frac {a b \int \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx}{2} - a \int x \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx - b \int x \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx + \int x^{2} \sqrt {\frac {1}{x \left (a b - a x - b x + x^{2}\right )}}\, dx\right ]
\]