77.1.55 problem 74 (page 112)

Internal problem ID [17945]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 74 (page 112)
Date solved : Tuesday, January 28, 2025 at 11:14:40 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.218 (sec). Leaf size: 325

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

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

✓ Solution by Mathematica

Time used: 1.302 (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*}