29.27.1 problem 766
Internal
problem
ID
[5351]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
27
Problem
number
:
766
Date
solved
:
Tuesday, March 04, 2025 at 09:31:27 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{2}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )&=0 \end{align*}
✓ Maple. Time used: 0.143 (sec). Leaf size: 112
ode:=diff(y(x),x)^2+f(x)*(y(x)-a)^2*(y(x)-b) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}+\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\frac {2 \arctan \left (\frac {\sqrt {y \left (x \right )-b}}{\sqrt {b -a}}\right )}{\sqrt {b -a}}-\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )}d \textit {\_a}}{\sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 60.102 (sec). Leaf size: 93
ode=(D[y[x],x])^2+f[x]*(y[x]-a)^2 *(y[x]-b)==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x-\sqrt {f(K[1])}dK[1]+c_1\right )\right ) \\
y(x)\to b+(b-a) \tan ^2\left (\frac {1}{2} \sqrt {a-b} \left (\int _1^x\sqrt {f(K[2])}dK[2]+c_1\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 6.595 (sec). Leaf size: 233
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
f = Function("f")
ode = Eq((-a + y(x))**2*(-b + y(x))*f(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - \begin {cases} \frac {y{\left (x \right )}}{a \sqrt {- b + y{\left (x \right )}} - \sqrt {- b + y{\left (x \right )}} y{\left (x \right )}} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\\frac {{G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{a \sqrt {- b + y{\left (x \right )}} - \sqrt {- b + y{\left (x \right )}} y{\left (x \right )}} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{a \sqrt {- b + y{\left (x \right )}} - \sqrt {- b + y{\left (x \right )}} y{\left (x \right )}} & \text {otherwise} \end {cases} - \frac {\int \sqrt {\left (b - y{\left (x \right )}\right ) f{\left (x \right )}}\, dx}{\sqrt {- b + y{\left (x \right )}}} = C_{1}, \ - \begin {cases} \frac {y{\left (x \right )}}{a \sqrt {- b + y{\left (x \right )}} - \sqrt {- b + y{\left (x \right )}} y{\left (x \right )}} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\\frac {{G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{a \sqrt {- b + y{\left (x \right )}} - \sqrt {- b + y{\left (x \right )}} y{\left (x \right )}} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{a \sqrt {- b + y{\left (x \right )}} - \sqrt {- b + y{\left (x \right )}} y{\left (x \right )}} & \text {otherwise} \end {cases} + \frac {\int \sqrt {\left (b - y{\left (x \right )}\right ) f{\left (x \right )}}\, dx}{\sqrt {- b + y{\left (x \right )}}} = C_{1}\right ]
\]