Internal problem ID [9428]
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: 0.203 (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 \relax (x ) = -\frac {i x^{2}}{2 b}+c_{1} x +c_{2} \\ y \relax (x ) = \frac {i x^{2}}{2 b}+c_{1} x +c_{2} \\ y \relax (x ) = \int \left (\frac {\sqrt {a^{2} b^{4} x^{2}+2 a^{2} b^{4} x c_{1}+a^{2} b^{4} c_{1}^{2}-1}\, x}{2 b}+\frac {\sqrt {a^{2} b^{4} x^{2}+2 a^{2} b^{4} x c_{1}+a^{2} b^{4} c_{1}^{2}-1}\, c_{1}}{2 b}-\frac {\ln \left (\frac {a^{2} b^{4} x +a^{2} b^{4} c_{1}}{\sqrt {b^{4} a^{2}}}+\sqrt {a^{2} b^{4} x^{2}+2 a^{2} b^{4} x c_{1}+a^{2} b^{4} c_{1}^{2}-1}\right )}{2 b \sqrt {b^{4} a^{2}}}\right )d x +x c_{2}+c_{3} \\ y \relax (x ) = \int \left (-\frac {\sqrt {a^{2} b^{4} x^{2}+2 a^{2} b^{4} x c_{1}+a^{2} b^{4} c_{1}^{2}-1}\, x}{2 b}-\frac {\sqrt {a^{2} b^{4} x^{2}+2 a^{2} b^{4} x c_{1}+a^{2} b^{4} c_{1}^{2}-1}\, c_{1}}{2 b}+\frac {\ln \left (\frac {a^{2} b^{4} x +a^{2} b^{4} c_{1}}{\sqrt {b^{4} a^{2}}}+\sqrt {a^{2} b^{4} x^{2}+2 a^{2} b^{4} x c_{1}+a^{2} b^{4} c_{1}^{2}-1}\right )}{2 b \sqrt {b^{4} a^{2}}}\right )d x +x c_{2}+c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 32.308 (sec). Leaf size: 379
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 {a^2 b^4 \left (x^2 \sqrt {-1+b^4 (a x+c_1){}^2}+6 b c_3 x+6 b c_2\right )+2 a b^4 c_1 x \sqrt {-1+b^4 (a x+c_1){}^2}+\left (2+b^4 c_1{}^2\right ) \sqrt {-1+b^4 (a x+c_1){}^2}-3 b^2 \left (c_1 \log \left (\sqrt {-1+b^4 (a x+c_1){}^2}+b^2 (a x+c_1)\right )+a x \log \left (b^4 (a x+c_1)+b^2 \sqrt {-1+b^4 (a x+c_1){}^2}\right )\right )}{6 a^2 b^5} \\ y(x)\to \frac {a^2 b^4 \left (x^2 \left (-\sqrt {-1+b^4 (a x+c_1){}^2}\right )+6 b c_3 x+6 b c_2\right )-2 a b^4 c_1 x \sqrt {-1+b^4 (a x+c_1){}^2}-\left (2+b^4 c_1{}^2\right ) \sqrt {-1+b^4 (a x+c_1){}^2}+3 b^2 \left (c_1 \log \left (\sqrt {-1+b^4 (a x+c_1){}^2}+b^2 (a x+c_1)\right )+a x \log \left (b^4 (a x+c_1)+b^2 \sqrt {-1+b^4 (a x+c_1){}^2}\right )\right )}{6 a^2 b^5} \\ \end{align*}