Internal problem ID [5131]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T.
CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page
218
Problem number: Problem 3.18.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
Solve \begin {gather*} \boxed {a y^{\prime \prime } y^{\prime \prime \prime }-\sqrt {1+\left (y^{\prime \prime }\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.218 (sec). Leaf size: 237
dsolve(a*diff(y(x),x$2)*diff(y(x),x$3)=sqrt(1+ diff(y(x),x$2)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) = \frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) = \frac {\left (-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}\right )^{\frac {3}{2}}}{6 a}-\frac {a \ln \left (c_{1} +x +\sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}\right ) x}{2}-\frac {a \ln \left (c_{1} +x +\sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}\right ) c_{1}}{2}+\frac {a \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}}{2}+x c_{2} +c_{3} \\ y \left (x \right ) = -\frac {\left (-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}\right )^{\frac {3}{2}}}{6 a}+\frac {a \ln \left (c_{1} +x +\sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}\right ) x}{2}+\frac {a \ln \left (c_{1} +x +\sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}\right ) c_{1}}{2}-\frac {a \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}}{2}+x c_{2} +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.255 (sec). Leaf size: 193
DSolve[a*y''[x]*y'''[x]==Sqrt[1+ y''[x]^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {(x+a (-1+c_1)) (a+a c_1+x)} \left (a^2 \left (2+c_1{}^2\right )+2 a c_1 x+x^2\right )}{6 a}-\frac {1}{2} a (x+a c_1) \log \left (\sqrt {(x+a (-1+c_1)) (a+a c_1+x)}+a c_1+x\right )+c_3 x+c_2 \\ y(x)\to -\frac {\sqrt {(x+a (-1+c_1)) (a+a c_1+x)} \left (a^2 \left (2+c_1{}^2\right )+2 a c_1 x+x^2\right )}{6 a}+\frac {1}{2} a (x+a c_1) \log \left (\sqrt {(x+a (-1+c_1)) (a+a c_1+x)}+a c_1+x\right )+c_3 x+c_2 \\ \end{align*}