15.19.22 problem 22
Internal
problem
ID
[3306]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
37,
page
171
Problem
number
:
22
Date
solved
:
Sunday, March 30, 2025 at 01:34:19 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left ({y^{\prime }}^{2}+1\right ) x&=y^{\prime } \left (x +y\right ) \end{align*}
✓ Maple. Time used: 0.035 (sec). Leaf size: 36
ode:=(1+diff(y(x),x)^2)*x = (x+y(x))*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= x \\
y &= \frac {x \left (\operatorname {LambertW}\left (\frac {x}{c_1}\right )^{2}-\operatorname {LambertW}\left (\frac {x}{c_1}\right )+1\right )}{\operatorname {LambertW}\left (\frac {x}{c_1}\right )} \\
\end{align*}
✓ Mathematica. Time used: 5.364 (sec). Leaf size: 247
ode=(D[y[x],x]^2+1)*x==D[y[x],x]*(x+y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [2 \left (\text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+3}-2}{\sqrt {\frac {y(x)}{x}-1}}\right )+\frac {\sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )}{\frac {2 y(x)}{x}-2 \sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )-4 \sqrt {\frac {y(x)}{x}+3}+6}\right )&=\frac {\log (x)}{2}+c_1,y(x)\right ] \\
\text {Solve}\left [2 \text {arctanh}\left (\frac {\sqrt {\frac {y(x)}{x}+3}-2}{\sqrt {\frac {y(x)}{x}-1}}\right )+\frac {\sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )}{\frac {y(x)}{x}+\sqrt {\frac {y(x)}{x}-1} \left (\sqrt {\frac {y(x)}{x}+3}-2\right )-2 \sqrt {\frac {y(x)}{x}+3}+3}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 56.834 (sec). Leaf size: 185
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(Derivative(y(x), x)**2 + 1) - (x + y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (u_{1} - \sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1} - 1\right )}\, du_{1} + \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1}}{u_{1} \left (u_{1} - \sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1} - 1\right )}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} - \sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1} - 1}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (u_{1} + \sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1} - 1\right )}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1}}{u_{1} \left (u_{1} + \sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1} - 1\right )}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} + \sqrt {- 3 u_{1}^{2} + 2 u_{1} + 1} - 1}\, du_{1}}\right ]
\]