23.2.20 problem 20

Internal problem ID [5375]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 12:38:13 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 ) \left (y-b \right ) \left (y-c \right )&=0 \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 156

ode:=diff(y(x),x)^2+f(x)*(y(x)-a)*(y(x)-b)*(y(x)-c) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \int _{}^{y}\frac {1}{\sqrt {\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y-c \right ) \left (y-b \right ) \left (y-a \right )}d \textit {\_a}}{\sqrt {\left (y-c \right ) \left (y-b \right ) \left (y-a \right )}}+c_1 &= 0 \\ \int _{}^{y}\frac {1}{\sqrt {\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )}}d \textit {\_a} +\frac {\int _{}^{x}\sqrt {-f \left (\textit {\_a} \right ) \left (y-c \right ) \left (y-b \right ) \left (y-a \right )}d \textit {\_a}}{\sqrt {\left (y-c \right ) \left (y-b \right ) \left (y-a \right )}}+c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 19.781 (sec). Leaf size: 228

ode=(D[y[x],x])^2+f[x]*(y[x]-a)(y[x]-b)*(y[x]-c)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x-\sqrt {f(K[1])}dK[1]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x-\sqrt {f(K[1])}dK[1]\right )|\frac {a-c}{a-b}\right ){}^2-a+b\right )\\ y(x)&\to \text {ns}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x\sqrt {f(K[2])}dK[2]\right )|\frac {a-c}{a-b}\right ){}^2 \left (a \text {sn}\left (\frac {1}{2} \sqrt {a-b} \left (c_1+\int _1^x\sqrt {f(K[2])}dK[2]\right )|\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: 50.241 (sec). Leaf size: 541

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))*f(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]