29.27.4 problem 770
Internal
problem
ID
[5354]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
27
Problem
number
:
770
Date
solved
:
Sunday, March 30, 2025 at 08:02:38 AM
CAS
classification
:
[_separable]
\begin{align*} {y^{\prime }}^{2}&=f \left (x \right )^{2} \left (y-a \right ) \left (y-b \right ) \left (y-c \right )^{2} \end{align*}
✓ Maple. Time used: 0.587 (sec). Leaf size: 284
ode:=diff(y(x),x)^2 = f(x)^2*(y(x)-a)*(y(x)-b)*(y(x)-c)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {c \,{\mathrm e}^{2 \left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+\left (\left (4 b -2 c \right ) a -2 c b \right ) {\mathrm e}^{\left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+c \left (a -b \right )^{2}}{{\mathrm e}^{2 \left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+\left (2 a +2 b -4 c \right ) {\mathrm e}^{\left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+a^{2}-2 a b +b^{2}} \\
y &= \frac {\left (\left (4 b -2 c \right ) a -2 c b \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+c \left ({\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+\left (a -b \right )^{2}\right )}{\left (2 a +2 b -4 c \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}+a^{2}-2 a b +b^{2}+{\mathrm e}^{-2 \left (\int f \left (x \right )d x +c_1 \right ) \sqrt {\left (b -c \right ) \left (a -c \right )}}} \\
\end{align*}
✓ Mathematica. Time used: 60.72 (sec). Leaf size: 223
ode=(D[y[x],x])^2==f[x]^2 *(y[x]-a)*(y[x]-b)*(y[x]-c)^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {b (a-c)+a (b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^x-f(K[1])dK[1]+c_1\right )\right )}{(b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^x-f(K[1])dK[1]+c_1\right )\right )+a-c} \\
y(x)\to \frac {b (a-c)+a (b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^xf(K[2])dK[2]+c_1\right )\right )}{(b-c) \tan ^2\left (\frac {1}{2} \sqrt {c-a} \sqrt {b-c} \left (\int _1^xf(K[2])dK[2]+c_1\right )\right )+a-c} \\
\end{align*}
✓ Sympy. Time used: 14.311 (sec). Leaf size: 48
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
f = Function("f")
ode = Eq((a - y(x))*(-b + y(x))*(-c + y(x))**2*f(x)**2 + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\left (- y + a\right ) \left (- y + b\right )} \left (- y + c\right )}\, dy = C_{1} - \int f{\left (x \right )}\, dx, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\left (- y + a\right ) \left (- y + b\right )} \left (- y + c\right )}\, dy = C_{1} + \int f{\left (x \right )}\, dx\right ]
\]