Internal problem ID [3226]
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: 4.
CAS Maple gives this as type [_quadrature]
\[ \boxed {-y^{\prime } \sqrt {1+{y^{\prime }}^{2}}=-x} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 147
dsolve(x=diff(y(x),x)*sqrt( (diff(y(x),x))^2+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {i \left (-32 x^{4}-4 x^{2}+1\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3 \sqrt {4 x^{2}+1}}-\frac {16 i x^{3} \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3}+c_{1} \\ y \left (x \right ) &= \frac {i \left (-32 x^{4}-4 x^{2}+1\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3 \sqrt {4 x^{2}+1}}+\frac {16 i x^{3} \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3}+c_{1} \\ y \left (x \right ) &= -\frac {\left (\int \sqrt {2 \sqrt {4 x^{2}+1}-2}d x \right )}{2}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \sqrt {2 \sqrt {4 x^{2}+1}-2}d x \right )}{2}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.161 (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*}