Internal problem ID [10181]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1849.
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 {y^{\prime \prime } y^{\prime \prime \prime }-a \sqrt {\left (y^{\prime \prime }\right )^{2} b^{2}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 371
dsolve(diff(diff(y(x),x),x)*diff(diff(diff(y(x),x),x),x)-a*(b^2*diff(diff(y(x),x),x)^2+1)^(1/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {i x^{2}}{2 b}+x c_{1} +c_{2} y \left (x \right ) = \frac {i x^{2}}{2 b}+x c_{1} +c_{2} y \left (x \right ) = \int \left (\frac {\sqrt {c_{1}^{2} a^{2} b^{4}+2 c_{1} a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, x}{2 b}+\frac {\sqrt {c_{1}^{2} a^{2} b^{4}+2 c_{1} a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, c_{1}}{2 b}-\frac {\ln \left (\frac {c_{1} a^{2} b^{4}+a^{2} b^{4} x}{\sqrt {b^{4} a^{2}}}+\sqrt {c_{1}^{2} a^{2} b^{4}+2 c_{1} a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\right )}{2 b \sqrt {b^{4} a^{2}}}\right )d x +c_{2} x +c_{3} y \left (x \right ) = \int \left (-\frac {\sqrt {c_{1}^{2} a^{2} b^{4}+2 c_{1} a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, x}{2 b}-\frac {\sqrt {c_{1}^{2} a^{2} b^{4}+2 c_{1} a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\, c_{1}}{2 b}+\frac {\ln \left (\frac {c_{1} a^{2} b^{4}+a^{2} b^{4} x}{\sqrt {b^{4} a^{2}}}+\sqrt {c_{1}^{2} a^{2} b^{4}+2 c_{1} a^{2} b^{4} x +a^{2} b^{4} x^{2}-1}\right )}{2 b \sqrt {b^{4} a^{2}}}\right )d x +c_{2} x +c_{3} \end{align*}
✓ Solution by Mathematica
Time used: 31.226 (sec). Leaf size: 415
DSolve[-(a*Sqrt[1 + b^2*y''[x]^2]) + y''[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {6 a^2 b^5 c_3 x+6 a^2 b^5 c_2+\left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1\right ){}^{3/2}+3 \sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}-3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )-3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5} y(x)\to \frac {-\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1} \left (a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2+2\right )+3 b^2 c_1 \log \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )+3 a b^2 x \log \left (b^2 \left (\sqrt {a^2 b^4 x^2+2 a b^4 c_1 x+b^4 c_1{}^2-1}+a b^2 x+b^2 c_1\right )\right )}{6 a^2 b^5}+c_3 x+c_2 \end{align*}