9.20.6 problem 6
Internal
problem
ID
[3314]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
38,
page
173
Problem
number
:
6
Date
solved
:
Tuesday, September 30, 2025 at 06:35:55 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left ({y^{\prime }}^{2}+1\right ) y&=2 y^{\prime } x \end{align*}
✓ Maple. Time used: 0.054 (sec). Leaf size: 67
ode:=y(x)*(1+diff(y(x),x)^2) = 2*x*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -x \\
y &= x \\
y &= 0 \\
y &= \sqrt {c_1 \left (-2 i x +c_1 \right )} \\
y &= \sqrt {c_1 \left (2 i x +c_1 \right )} \\
y &= -\sqrt {c_1 \left (-2 i x +c_1 \right )} \\
y &= -\sqrt {c_1 \left (2 i x +c_1 \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.636 (sec). Leaf size: 64
ode=(D[y[x],x]^2+1)*y[x]==2*D[y[x],x]*x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )}\\ y(x)&\to \sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )}\\ y(x)&\to 0\\ y(x)&\to -x\\ y(x)&\to x \end{align*}
✓ Sympy. Time used: 158.131 (sec). Leaf size: 469
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*Derivative(y(x), x) + (Derivative(y(x), x)**2 + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}\right ]
\]