60.2.15 problem 591

Internal problem ID [10589]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 591
Date solved : Sunday, March 30, 2025 at 06:08:55 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {F \left (\frac {a y^{2}+b \,x^{2}}{a}\right ) x}{\sqrt {a}\, y} \end{align*}

✓ Maple. Time used: 0.167 (sec). Leaf size: 126

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

\begin{align*} y &= \operatorname {RootOf}\left (F \left (\frac {a \,\textit {\_Z}^{2}+b \,x^{2}}{a}\right ) \sqrt {a}+b \right ) \\ y &= \frac {\sqrt {a \left (-b \,x^{2}+\operatorname {RootOf}\left (a \int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) \sqrt {a}+b}d \textit {\_a} b -b \,x^{2}+2 c_1 a \right ) a \right )}}{a} \\ y &= -\frac {\sqrt {a \left (-b \,x^{2}+\operatorname {RootOf}\left (a \int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) \sqrt {a}+b}d \textit {\_a} b -b \,x^{2}+2 c_1 a \right ) a \right )}}{a} \\ \end{align*}

✓ Mathematica. Time used: 0.434 (sec). Leaf size: 253

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {b K[2]}{b+\sqrt {a} F\left (\frac {b x^2+a K[2]^2}{a}\right )}-\int _1^x\left (\frac {2 b K[1] K[2] F''\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )}-\frac {2 b F\left (\frac {b K[1]^2+a K[2]^2}{a}\right ) K[1] K[2] F''\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b F\left (\frac {b K[1]^2+a y(x)^2}{a}\right ) K[1]}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )\right )}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - x*F(y(x)**2 + b*x**2/a)/(sqrt(a)*y(x)) cannot be solved by the factorable group method