66.1.39 problem Problem 53

Internal problem ID [13886]
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 : Tuesday, January 28, 2025 at 06:07:25 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.096 (sec). Leaf size: 339

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

\begin{align*} y &= a \\ y &= \frac {\left (\operatorname {RootOf}\left (\left (a \cos \left (\textit {\_Z} \right )+a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+a \textit {\_Z} +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 )+a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+a \textit {\_Z} +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 )+a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+a \textit {\_Z} +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 )+a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )+a \textit {\_Z} +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 )-a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-a \textit {\_Z} +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 )-a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-a \textit {\_Z} +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 )-a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-a \textit {\_Z} +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 )-a \textit {\_Z} +2 c_{1} -2 x \right ) \left (-a \cos \left (\textit {\_Z} \right )-a \textit {\_Z} +2 c_{1} -2 x \right )\right )\right )}{2}+\frac {a}{2} \\ \end{align*}

✓ Solution by Mathematica

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

DSolve[y[x]*(1+D[y[x],x]^2)==a,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*}