31.5.6 problem 12

Internal problem ID [5749]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 6
Problem number : 12
Date solved : Sunday, March 30, 2025 at 10:07:29 AM
CAS classification : [_separable]

\begin{align*} \frac {\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}\, y^{\prime }}{\sqrt {a +b y+c y^{2}+c y^{3}+f y^{4}}}&=-1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 56

ode:=(f*x^4+c*x^3+c*x^2+b*x+a)^(1/2)/(a+b*y(x)+c*y(x)^2+c*y(x)^3+f*y(x)^4)^(1/2)*diff(y(x),x) = -1; 
dsolve(ode,y(x), singsol=all);
 

\[ \int \frac {1}{\sqrt {f \,x^{4}+c \,x^{3}+c \,x^{2}+b x +a}}d x +\int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{4} f +\textit {\_a}^{3} c +\textit {\_a}^{2} c +\textit {\_a} b +a}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 21.58 (sec). Leaf size: 2239

ode=Sqrt[a+b*x+c*x^2+c*x^3+f*x^4]/Sqrt[a+b*y[x]+c*y[x]^2+c*y[x]^3+f*y[x]^4]*D[y[x],x]==-1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 1.197 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
f = symbols("f") 
y = Function("y") 
ode = Eq(sqrt(a + b*x + c*x**3 + c*x**2 + f*x**4)*Derivative(y(x), x)/sqrt(a + b*y(x) + c*y(x)**3 + c*y(x)**2 + f*y(x)**4) + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {y^{4} f + y^{3} c + y^{2} c + y b + a}}\, dy = C_{1} - \int \frac {1}{\sqrt {a + b x + c x^{3} + c x^{2} + f x^{4}}}\, dx \]