Internal problem ID [2717]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 84.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {x -y^{\prime } \sqrt {\left (y^{\prime }\right )^{2}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.075 (sec). Leaf size: 187
dsolve(x=diff(y(x),x)*sqrt( (diff(y(x),x))^2+1),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {i \sqrt {2}\, \left (-\frac {256 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \left (2 x \right )}{2}\right )}{3}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {64}{3} x^{4}-\frac {8}{3} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (2 x \right )}{2}\right )}{\sqrt {4 x^{2}+1}}\right )}{32 \sqrt {\pi }}+c_{1} \\ y \relax (x ) = -\frac {i \sqrt {2}\, \left (-\frac {256 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \left (2 x \right )}{2}\right )}{3}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {64}{3} x^{4}-\frac {8}{3} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (2 x \right )}{2}\right )}{\sqrt {4 x^{2}+1}}\right )}{32 \sqrt {\pi }}+c_{1} \\ y \relax (x ) = \int -\frac {\sqrt {-2+2 \sqrt {4 x^{2}+1}}}{2}d x +c_{1} \\ y \relax (x ) = \int \frac {\sqrt {-2+2 \sqrt {4 x^{2}+1}}}{2}d x +c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.146 (sec). Leaf size: 207
DSolve[x==y'[x]*Sqrt[ (y'[x])^2+1],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {2} x \left (\sqrt {4 x^2+1}-2\right )}{3 \sqrt {\sqrt {4 x^2+1}-1}}+c_1 \\ y(x)\to \frac {\sqrt {2} x \left (\sqrt {4 x^2+1}-2\right )}{3 \sqrt {\sqrt {4 x^2+1}-1}}+c_1 \\ y(x)\to -\frac {\sqrt {2} x \left (4 x^2+3 \sqrt {4 x^2+1}+3\right )}{3 \left (-\sqrt {4 x^2+1}-1\right )^{3/2}}+c_1 \\ y(x)\to \frac {\sqrt {2} x \left (4 x^2+3 \sqrt {4 x^2+1}+3\right )}{3 \left (-\sqrt {4 x^2+1}-1\right )^{3/2}}+c_1 \\ \end{align*}