60.1.63 problem 64

Internal problem ID [10077]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 64
Date solved : Sunday, March 30, 2025 at 03:06:59 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 153

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

\[ -\frac {\sqrt {a \,x^{2}+b x +c}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{\sqrt {a}}\right )\right ) \sqrt {\frac {a y^{2}+b y+c}{a \,x^{2}+b x +c}}-\left (\sqrt {a}\, c_1 +\ln \left (\frac {2 \sqrt {a y^{2}+b y+c}\, \sqrt {a}+2 a y+b}{\sqrt {a}}\right )-\ln \left (2\right )\right ) \sqrt {a y^{2}+b y+c}}{\sqrt {a}\, \sqrt {a y^{2}+b y+c}} = 0 \]

✓ Mathematica. Time used: 5.601 (sec). Leaf size: 142

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

\begin{align*} y(x)\to \frac {e^{-\sqrt {a} c_1} \left (2 \sqrt {a} \left (-1+e^{2 \sqrt {a} c_1}\right ) \sqrt {x (a x+b)+c}+b \left (-1+e^{\sqrt {a} c_1}\right ){}^2+2 a x \left (1+e^{2 \sqrt {a} c_1}\right )\right )}{4 a} \\ y(x)\to -\frac {\sqrt {b^2-4 a c}+b}{2 a} \\ y(x)\to \frac {\sqrt {b^2-4 a c}-b}{2 a} \\ \end{align*}

✗ Sympy

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

Timed Out