60.1.405 problem 416
Internal
problem
ID
[10419]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
416
Date
solved
:
Sunday, March 30, 2025 at 04:41:17 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}+\left (y-3 x \right ) y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.039 (sec). Leaf size: 140
ode:=x*diff(y(x),x)^2+(y(x)-3*x)*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= x \\
-\frac {c_1 \left (5 x -y+\sqrt {9 x^{2}-10 y x +y^{2}}\right )}{x {\left (\frac {3 x -y+\sqrt {9 x^{2}-10 y x +y^{2}}}{x}\right )}^{{3}/{2}}}+x &= 0 \\
\frac {\left (-5 x +y+\sqrt {9 x^{2}-10 y x +y^{2}}\right ) c_1 \sqrt {2}}{4 x {\left (\frac {-y+3 x -\sqrt {9 x^{2}-10 y x +y^{2}}}{x}\right )}^{{3}/{2}}}+x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 60.13 (sec). Leaf size: 1221
ode=y[x] + (-3*x + y[x])*D[y[x],x] + x*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 53.587 (sec). Leaf size: 197
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x)**2 + (-3*x + y(x))*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{\int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (3 u_{1} - \sqrt {9 u_{1}^{2} - 10 u_{1} + 1} - 3\right )}\, du_{1} + \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {9 u_{1}^{2} - 10 u_{1} + 1}}{u_{1} \left (3 u_{1} - \sqrt {9 u_{1}^{2} - 10 u_{1} + 1} - 3\right )}\, du_{1} - 3 \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{3 u_{1} - \sqrt {9 u_{1}^{2} - 10 u_{1} + 1} - 3}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{\int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (3 u_{1} + \sqrt {9 u_{1}^{2} - 10 u_{1} + 1} - 3\right )}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {9 u_{1}^{2} - 10 u_{1} + 1}}{u_{1} \left (3 u_{1} + \sqrt {9 u_{1}^{2} - 10 u_{1} + 1} - 3\right )}\, du_{1} - 3 \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{3 u_{1} + \sqrt {9 u_{1}^{2} - 10 u_{1} + 1} - 3}\, du_{1}}\right ]
\]