89.32.24 problem 27
Internal
problem
ID
[24984]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
16.
Equations
of
order
one
and
higher
degree.
Miscellaneous
Exercises
at
page
246
Problem
number
:
27
Date
solved
:
Thursday, October 02, 2025 at 11:45:53 PM
CAS
classification
:
[_dAlembert]
\begin{align*} {y^{\prime }}^{2}+y y^{\prime }-x -1&=0 \end{align*}
✓ Maple. Time used: 0.133 (sec). Leaf size: 661
ode:=diff(y(x),x)^2+y(x)*diff(y(x),x)-x-1 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {-\sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}-4}\, \sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}+4}\, \left (y+\sqrt {y^{2}+4 x +4}\right ) \ln \left (-y-\sqrt {y^{2}+4 x +4}+\sqrt {2 y^{2}+2 y \sqrt {y^{2}+4 x +4}+4 x}\right )+\left (\sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}+4}\, \left (1+x \right ) \sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}-4}+2 c_1 \left (y+\sqrt {y^{2}+4 x +4}\right )\right ) \sqrt {2 y^{2}+2 y \sqrt {y^{2}+4 x +4}+4 x}+\ln \left (2\right ) \sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}-4}\, \sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}+4}\, \left (y+\sqrt {y^{2}+4 x +4}\right )}{\sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}-4}\, \sqrt {2 y^{2}+2 y \sqrt {y^{2}+4 x +4}+4 x}\, \sqrt {-2 y-2 \sqrt {y^{2}+4 x +4}+4}} &= 0 \\
\frac {-\sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}-4}\, \sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}+4}\, \left (y-\sqrt {y^{2}+4 x +4}\right ) \ln \left (-y+\sqrt {y^{2}+4 x +4}+\sqrt {2 y^{2}-2 y \sqrt {y^{2}+4 x +4}+4 x}\right )+\left (\sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}+4}\, \left (1+x \right ) \sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}-4}+2 c_1 \left (y-\sqrt {y^{2}+4 x +4}\right )\right ) \sqrt {2 y^{2}-2 y \sqrt {y^{2}+4 x +4}+4 x}+\ln \left (2\right ) \sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}-4}\, \sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}+4}\, \left (y-\sqrt {y^{2}+4 x +4}\right )}{\sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}-4}\, \sqrt {2 y^{2}-2 y \sqrt {y^{2}+4 x +4}+4 x}\, \sqrt {-2 y+2 \sqrt {y^{2}+4 x +4}+4}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.317 (sec). Leaf size: 106
ode=D[y[x],x]^2+y[x]*D[y[x],x]-x-1==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left \{x=\frac {K[1] \left (-2 \arctan \left (\frac {\sqrt {1-K[1]^2}}{K[1]+1}\right )-\frac {\sqrt {1-K[1]^2}}{K[1]}\right )}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}+\frac {1-K[1]^2}{K[1]}\right \},\{y(x),K[1]\}\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + y(x)*Derivative(y(x), x) + Derivative(y(x), x)**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out