9.19.13 problem 13
Internal
problem
ID
[3297]
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
:
13
Date
solved
:
Tuesday, September 30, 2025 at 06:32:57 AM
CAS
classification
:
[_quadrature]
\begin{align*} y \left (1+{y^{\prime }}^{2}\right )&=2 \end{align*}
✓ Maple. Time used: 0.045 (sec). Leaf size: 50
ode:=y(x)*(1+diff(y(x),x)^2) = 2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 2 \\
y &= -\sin \left (\operatorname {RootOf}\left (-\textit {\_Z} -x -\operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+c_1 \right )\right )+1 \\
y &= \sin \left (\operatorname {RootOf}\left (-\textit {\_Z} -x +\operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+c_1 \right )\right )+1 \\
\end{align*}
✓ Mathematica. Time used: 0.189 (sec). Leaf size: 98
ode=y[x]*(1+D[y[x],x]^2)==2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-2 \arctan \left (\frac {\sqrt {2-\text {$\#$1}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {-((\text {$\#$1}-2) \text {$\#$1})}\&\right ][-x+c_1]\\ y(x)&\to \text {InverseFunction}\left [-2 \arctan \left (\frac {\sqrt {2-\text {$\#$1}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {-((\text {$\#$1}-2) \text {$\#$1})}\&\right ][x+c_1]\\ y(x)&\to 2 \end{align*}
✓ Sympy. Time used: 1.326 (sec). Leaf size: 218
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((Derivative(y(x), x)**2 + 1)*y(x) - 2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2} \right )} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 2}} + \frac {2 i \sqrt {y{\left (x \right )}}}{\sqrt {y{\left (x \right )} - 2}} & \text {for}\: \left |{y{\left (x \right )}}\right | > 2 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {2 - y{\left (x \right )}}} - \frac {2 \sqrt {y{\left (x \right )}}}{\sqrt {2 - y{\left (x \right )}}} & \text {otherwise} \end {cases} = C_{1} - x, \ \begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2} \right )} - \frac {i y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {y{\left (x \right )} - 2}} + \frac {2 i \sqrt {y{\left (x \right )}}}{\sqrt {y{\left (x \right )} - 2}} & \text {for}\: \left |{y{\left (x \right )}}\right | > 2 \\2 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {y{\left (x \right )}}}{2} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {2 - y{\left (x \right )}}} - \frac {2 \sqrt {y{\left (x \right )}}}{\sqrt {2 - y{\left (x \right )}}} & \text {otherwise} \end {cases} = C_{1} + x\right ]
\]