29.26.26 problem 762

Internal problem ID [5347]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 762
Date solved : Tuesday, March 04, 2025 at 09:30:11 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=\left (y-a \right ) \left (y-b \right ) \left (y-c \right ) \end{align*}

✓ Maple. Time used: 0.046 (sec). Leaf size: 75

ode:=diff(y(x),x)^2 = (y(x)-a)*(y(x)-b)*(y(x)-c); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ y \left (x \right ) &= c \\ x -\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} -c_{1} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\left (\textit {\_a} -a \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -c \right )}}d \textit {\_a} -c_{1} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 33.309 (sec). Leaf size: 188

ode=(D[y[x],x])^2==(y[x]-a)(y[x]-b)*(y[x]-c); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} (c_1-i x)|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} (c_1-i x)|\frac {a-c}{a-b}\right ){}^2-a+b\right ) \\ y(x)\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} (i x+c_1)|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} (i x+c_1)|\frac {a-c}{a-b}\right ){}^2-a+b\right ) \\ y(x)\to a \\ y(x)\to b \\ y(x)\to c \\ \end{align*}

✓ Sympy. Time used: 16.177 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq((a - y(x))*(-b + y(x))*(-c + y(x)) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \left (- y + a\right ) \left (- y + b\right ) \left (- y + c\right )}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {- \left (- y + a\right ) \left (- y + b\right ) \left (- y + c\right )}}\, dy = C_{1} + x\right ] \]