54.1.459 problem 472
Internal
problem
ID
[11773]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
472
Date
solved
:
Tuesday, September 30, 2025 at 10:21:16 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (x +y\right ) {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.107 (sec). Leaf size: 121
ode:=(x+y(x))*diff(y(x),x)^2+2*x*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\left (1+i \sqrt {3}\right ) x}{2} \\
y &= \frac {\left (i \sqrt {3}-1\right ) x}{2} \\
\ln \left (x \right )-\operatorname {arctanh}\left (\frac {2 x +y}{2 x \sqrt {\frac {x^{2}+y x +y^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_1 &= 0 \\
\ln \left (x \right )+\operatorname {arctanh}\left (\frac {2 x +y}{2 x \sqrt {\frac {x^{2}+y x +y^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 2.157 (sec). Leaf size: 166
ode=-y[x] + 2*x*D[y[x],x] + (x + y[x])*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {2}{3} \sqrt {e^{c_1} \left (-3 x+e^{c_1}\right )}-\frac {e^{c_1}}{3}\\ y(x)&\to \frac {2}{3} \sqrt {e^{c_1} \left (-3 x+e^{c_1}\right )}-\frac {e^{c_1}}{3}\\ y(x)&\to e^{c_1}-2 \sqrt {e^{c_1} \left (x+e^{c_1}\right )}\\ y(x)&\to 2 \sqrt {e^{c_1} \left (x+e^{c_1}\right )}+e^{c_1}\\ y(x)&\to 0\\ y(x)&\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) x\\ y(x)&\to \frac {1}{2} i \left (\sqrt {3}+i\right ) x \end{align*}
✓ Sympy. Time used: 12.874 (sec). Leaf size: 131
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) + (x + y(x))*Derivative(y(x), x)**2 - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1}}{u_{1}^{2} + u_{1} \sqrt {u_{1}^{2} + u_{1} + 1} + u_{1} + 1}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {u_{1}^{2} + u_{1} + 1}}{u_{1}^{2} + u_{1} \sqrt {u_{1}^{2} + u_{1} + 1} + u_{1} + 1}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1}}{u_{1}^{2} - u_{1} \sqrt {u_{1}^{2} + u_{1} + 1} + u_{1} + 1}\, du_{1} + \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {u_{1}^{2} + u_{1} + 1}}{u_{1}^{2} - u_{1} \sqrt {u_{1}^{2} + u_{1} + 1} + u_{1} + 1}\, du_{1}}\right ]
\]