54.1.306 problem 312

Internal problem ID [11620]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 312
Date solved : Sunday, October 12, 2025 at 01:48:54 AM
CAS classification : [_rational]

\begin{align*} \left (\frac {y^{2}}{b}+\frac {x^{2}}{a}\right ) \left (y y^{\prime }+x \right )+\frac {\left (a -b \right ) \left (y y^{\prime }-x \right )}{a +b}&=0 \end{align*}

✓ Maple. Time used: 0.725 (sec). Leaf size: 244

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

\begin{align*} y &= \frac {\sqrt {\left ({\mathrm e}^{\frac {-2 \operatorname {LambertW}\left (\frac {\left (a +b \right ) {\mathrm e}^{\frac {\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_1 \right ) a +b^{2} x^{2}}{2 a^{2} b}}}{2 a^{2} b}\right ) a^{2} b +\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_1 \right ) a +b^{2} x^{2}}{2 a^{2} b}}+b \left (-x^{2}+a \right )\right ) a}}{a} \\ y &= -\frac {\sqrt {\left ({\mathrm e}^{\frac {-2 \operatorname {LambertW}\left (\frac {\left (a +b \right ) {\mathrm e}^{\frac {\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_1 \right ) a +b^{2} x^{2}}{2 a^{2} b}}}{2 a^{2} b}\right ) a^{2} b +\left (-x^{2}-b \right ) a^{2}+\left (-b^{2}-2 c_1 \right ) a +b^{2} x^{2}}{2 a^{2} b}}+b \left (-x^{2}+a \right )\right ) a}}{a} \\ \end{align*}

✓ Mathematica. Time used: 60.582 (sec). Leaf size: 190

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

\begin{align*} y(x)&\to -\frac {\sqrt {b} \sqrt {(a+b) \left (a-x^2\right )+2 a^2 W\left (\frac {c_1 (a+b) e^{-\frac {a^2 \left (b+x^2\right )+a b^2-b^2 x^2}{2 a^2 b}}}{2 a^3 b^2}\right )}}{\sqrt {a} \sqrt {a+b}}\\ y(x)&\to \frac {\sqrt {b} \sqrt {(a+b) \left (a-x^2\right )+2 a^2 W\left (\frac {c_1 (a+b) e^{-\frac {a^2 \left (b+x^2\right )+a b^2-b^2 x^2}{2 a^2 b}}}{2 a^3 b^2}\right )}}{\sqrt {a} \sqrt {a+b}} \end{align*}

✗ Sympy

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

Timed Out