60.1.306 problem 312
Internal
problem
ID
[10320]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
312
Date
solved
:
Sunday, March 30, 2025 at 04:09:23 PM
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: 60.949 (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