23.1.419 problem 408

Internal problem ID [5026]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 408
Date solved : Tuesday, September 30, 2025 at 11:27:22 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {b \,x^{4}+a \,x^{2}+1}+\sqrt {1+a y^{2}+b y^{4}}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 40

ode:=diff(y(x),x)*(b*x^4+a*x^2+1)^(1/2)+(1+a*y(x)^2+b*y(x)^4)^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 46.859 (sec). Leaf size: 505

ode=D[y[x],x]*Sqrt[1+a*x^2+b*x^4]+Sqrt[1+a*y[x]^2+b*y[x]^4]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {i \sqrt {\frac {2 \text {$\#$1}^2 b+\sqrt {a^2-4 b}+a}{\sqrt {a^2-4 b}+a}} \sqrt {\frac {2 \text {$\#$1}^2 b}{a-\sqrt {a^2-4 b}}+1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {b}{a+\sqrt {a^2-4 b}}} \text {$\#$1}\right ),\frac {a+\sqrt {a^2-4 b}}{a-\sqrt {a^2-4 b}}\right )}{\sqrt {2} \sqrt {\frac {b}{\sqrt {a^2-4 b}+a}} \sqrt {\text {$\#$1}^4 b+\text {$\#$1}^2 a+1}}\&\right ]\left [\frac {i \sqrt {\frac {\sqrt {a^2-4 b}+a+2 b x^2}{\sqrt {a^2-4 b}+a}} \sqrt {\frac {2 b x^2}{a-\sqrt {a^2-4 b}}+1} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {b}{a+\sqrt {a^2-4 b}}} x\right ),\frac {a+\sqrt {a^2-4 b}}{a-\sqrt {a^2-4 b}}\right )}{\sqrt {2} \sqrt {\frac {b}{\sqrt {a^2-4 b}+a}} \sqrt {a x^2+b x^4+1}}+c_1\right ]\\ y(x)&\to -\frac {\sqrt {-\frac {\sqrt {a^2-4 b}+a}{b}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {-\frac {\sqrt {a^2-4 b}+a}{b}}}{\sqrt {2}}\\ y(x)&\to -\frac {\sqrt {\frac {\sqrt {a^2-4 b}-a}{b}}}{\sqrt {2}}\\ y(x)&\to \frac {\sqrt {\frac {\sqrt {a^2-4 b}-a}{b}}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 0.504 (sec). Leaf size: 39

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

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