56.1.78 problem 78
Internal
problem
ID
[8790]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
78
Date
solved
:
Wednesday, March 05, 2025 at 06:48:41 AM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} \frac {y^{\prime } y}{1+\frac {\sqrt {1+{y^{\prime }}^{2}}}{2}}&=-x \end{align*}
✓ Maple. Time used: 1.737 (sec). Leaf size: 187
ode:=diff(y(x),x)*y(x)/(1+1/2*(1+diff(y(x),x)^2)^(1/2)) = -x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\sqrt {-x^{2}+c_{1}}\, \left (2+\sqrt {\frac {c_{1}}{-x^{2}+c_{1}}}\right )}{2} \\
y &= \frac {\sqrt {-x^{2}+c_{1}}\, \left (2+\sqrt {\frac {c_{1}}{-x^{2}+c_{1}}}\right )}{2} \\
y &= -\frac {\sqrt {-9 x^{2}+15 c_{1} -6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\
y &= \frac {\sqrt {-9 x^{2}+15 c_{1} -6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\
y &= -\frac {\sqrt {-9 x^{2}+15 c_{1} +6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\
y &= \frac {\sqrt {-9 x^{2}+15 c_{1} +6 \sqrt {-3 c_{1} x^{2}+4 c_{1}^{2}}}}{3} \\
\end{align*}
✓ Mathematica. Time used: 2.152 (sec). Leaf size: 153
ode=D[y[x],x]*y[x]/(1+1/2*Sqrt[1+(D[y[x],x])^2])==-x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{3} \left (e^{c_1}-\sqrt {-9 x^2+4 e^{2 c_1}}\right ) \\
y(x)\to \frac {1}{3} \left (\sqrt {-9 x^2+4 e^{2 c_1}}+e^{c_1}\right ) \\
y(x)\to -\sqrt {-x^2+4 e^{2 c_1}}-e^{c_1} \\
y(x)\to \sqrt {-x^2+4 e^{2 c_1}}-e^{c_1} \\
y(x)\to -\sqrt {-x^2} \\
y(x)\to \sqrt {-x^2} \\
\end{align*}
✓ Sympy. Time used: 85.557 (sec). Leaf size: 90
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x + y(x)*Derivative(y(x), x)/(sqrt(Derivative(y(x), x)**2 + 1)/2 + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {- 9 x^{2} + e^{2 C_{1}}}}{3} + \frac {e^{C_{1}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {- 9 x^{2} + e^{2 C_{1}}}}{3} + \frac {e^{C_{1}}}{6}, \ y{\left (x \right )} = - \frac {\sqrt {- 9 x^{2} + e^{2 C_{1}}}}{3} - \frac {e^{C_{1}}}{6}, \ y{\left (x \right )} = \frac {\sqrt {- 9 x^{2} + e^{2 C_{1}}}}{3} - \frac {e^{C_{1}}}{6}\right ]
\]