66.1.39 problem Problem 53

Internal problem ID [13815]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 53
Date solved : Monday, March 31, 2025 at 08:13:49 AM
CAS classification : [_quadrature]

\begin{align*} y \left (1+{y^{\prime }}^{2}\right )&=a \end{align*}

✓ Maple. Time used: 0.126 (sec). Leaf size: 339

ode:=y(x)*(1+diff(y(x),x)^2) = a; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= a \\ y &= \frac {\left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right ) a -2 x +2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ y &= \frac {\left (-\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right ) a +2 x -2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ y &= \frac {\left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right ) a +2 x -2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ y &= \frac {\left (-\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right ) a -2 x +2 c_1 \right ) \tan \left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-\textit {\_Z} a +2 c_1 -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ \end{align*}

✓ Mathematica. Time used: 1.256 (sec). Leaf size: 356

ode=y[x]*(1+D[y[x],x]^2)==a; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\left (\sqrt {\text {$\#$1}}-1\right ) \left (\sqrt {a-\text {$\#$1}}-\sqrt {a-1}\right ) \left (\text {$\#$1} a+\sqrt {a-1} a \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}} a-2 \sqrt {\text {$\#$1}} \sqrt {a-1} \sqrt {a-\text {$\#$1}}-2 \text {$\#$1}-a^2+a\right )}{\left (\sqrt {a-1} \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}}-a\right )^2}+2 a \arctan \left (\frac {1-\sqrt {\text {$\#$1}}}{\sqrt {a-1}-\sqrt {a-\text {$\#$1}}}\right )\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {\left (\sqrt {\text {$\#$1}}-1\right ) \left (\sqrt {a-\text {$\#$1}}-\sqrt {a-1}\right ) \left (\text {$\#$1} a+\sqrt {a-1} a \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}} a-2 \sqrt {\text {$\#$1}} \sqrt {a-1} \sqrt {a-\text {$\#$1}}-2 \text {$\#$1}-a^2+a\right )}{\left (\sqrt {a-1} \sqrt {a-\text {$\#$1}}+\sqrt {\text {$\#$1}}-a\right )^2}+2 a \arctan \left (\frac {1-\sqrt {\text {$\#$1}}}{\sqrt {a-1}-\sqrt {a-\text {$\#$1}}}\right )\&\right ][x+c_1] \\ y(x)\to a \\ \end{align*}

✓ Sympy. Time used: 1.664 (sec). Leaf size: 218

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a + (Derivative(y(x), x)**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \begin {cases} - i \sqrt {a} \sqrt {-1 + \frac {y{\left (x \right )}}{a}} \sqrt {y{\left (x \right )}} - i a \operatorname {acosh}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} & \text {for}\: \left |{\frac {y{\left (x \right )}}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {1 - \frac {y{\left (x \right )}}{a}}} + a \operatorname {asin}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {a} \sqrt {1 - \frac {y{\left (x \right )}}{a}}} & \text {otherwise} \end {cases} = C_{1} - x, \ \begin {cases} - i \sqrt {a} \sqrt {-1 + \frac {y{\left (x \right )}}{a}} \sqrt {y{\left (x \right )}} - i a \operatorname {acosh}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} & \text {for}\: \left |{\frac {y{\left (x \right )}}{a}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {y{\left (x \right )}}}{\sqrt {1 - \frac {y{\left (x \right )}}{a}}} + a \operatorname {asin}{\left (\frac {\sqrt {y{\left (x \right )}}}{\sqrt {a}} \right )} + \frac {y^{\frac {3}{2}}{\left (x \right )}}{\sqrt {a} \sqrt {1 - \frac {y{\left (x \right )}}{a}}} & \text {otherwise} \end {cases} = C_{1} + x\right ] \]