15.19.13 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 : Monday, January 27, 2025 at 07:31:42 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 50

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

\begin{align*} y \left (x \right ) &= 2 \\ y \left (x \right ) &= -\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 \left (x \right ) &= \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*}

✓ Solution by Mathematica

Time used: 1.422 (sec). Leaf size: 174

DSolve[y[x]*(1+D[y[x],x]^2)==2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [4 \arctan \left (\frac {\sqrt {\text {$\#$1}}-1}{\sqrt {2-\text {$\#$1}}-1}\right )-\frac {2 \left (\sqrt {2-\text {$\#$1}}-1\right )^2 \left (\sqrt {\text {$\#$1}}-1\right )^2}{\left (\sqrt {2-\text {$\#$1}}+\sqrt {\text {$\#$1}}-2\right )^2}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [4 \arctan \left (\frac {\sqrt {\text {$\#$1}}-1}{\sqrt {2-\text {$\#$1}}-1}\right )-\frac {2 \left (\sqrt {2-\text {$\#$1}}-1\right )^2 \left (\sqrt {\text {$\#$1}}-1\right )^2}{\left (\sqrt {2-\text {$\#$1}}+\sqrt {\text {$\#$1}}-2\right )^2}\&\right ][x+c_1] \\ y(x)\to 2 \\ \end{align*}