29.26.29 problem 765
Internal
problem
ID
[5350]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
765
Date
solved
:
Sunday, March 30, 2025 at 08:02:25 AM
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 ) \left (y-b \right )&=0 \end{align*}
✓ Maple. Time used: 0.036 (sec). Leaf size: 187
ode:=diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {\left (\ln \left (-a -b +2 y+2 \sqrt {\left (b -y\right ) \left (-y+a \right )}\right )+c_1 -\ln \left (2\right )\right ) \sqrt {\left (b -y\right ) \left (-y+a \right )}-\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y-b \right ) \left (y-a \right )}d \textit {\_a}}{\sqrt {\left (b -y\right ) \left (-y+a \right )}} &= 0 \\
\frac {\left (\ln \left (-a -b +2 y+2 \sqrt {\left (b -y\right ) \left (-y+a \right )}\right )+c_1 -\ln \left (2\right )\right ) \sqrt {\left (b -y\right ) \left (-y+a \right )}+\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y-b \right ) \left (y-a \right )}d \textit {\_a}}{\sqrt {\left (b -y\right ) \left (-y+a \right )}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 4.224 (sec). Leaf size: 89
ode=(D[y[x],x])^2+ f[x]*(y[x]-a)*(y[x]-b)==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{2} \left ((b-a) \cosh \left (\int _1^x-i \sqrt {f(K[2])}dK[2]+c_1\right )+a+b\right ) \\
y(x)\to \frac {1}{2} \left ((b-a) \cosh \left (\int _1^xi \sqrt {f(K[3])}dK[3]+c_1\right )+a+b\right ) \\
y(x)\to a \\
y(x)\to b \\
\end{align*}
✓ Sympy. Time used: 20.840 (sec). Leaf size: 272
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
f = Function("f")
ode = Eq((-a + y(x))*(-b + y(x))*f(x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} \frac {y{\left (x \right )}}{\sqrt {a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}}} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\\frac {{G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}}} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}}} & \text {otherwise} \end {cases} + \frac {\int \sqrt {- \left (a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}\right ) f{\left (x \right )}}\, dx}{\sqrt {\left (a - y{\left (x \right )}\right ) \left (b - y{\left (x \right )}\right )}} = C_{1}, \ \begin {cases} \frac {y{\left (x \right )}}{\sqrt {a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}}} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\\frac {{G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}}} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}}} & \text {otherwise} \end {cases} - \frac {\int \sqrt {- \left (a b - a y{\left (x \right )} - b y{\left (x \right )} + y^{2}{\left (x \right )}\right ) f{\left (x \right )}}\, dx}{\sqrt {\left (a - y{\left (x \right )}\right ) \left (b - y{\left (x \right )}\right )}} = C_{1}\right ]
\]