Internal problem ID [9427]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 7, non-linear third and higher order
Problem number: 1848.
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], [_3rd_order, _reducible, _mu_poly_yn]]
Solve \begin {gather*} \boxed {\left (\left (y^{\prime }\right )^{2}+1\right ) y^{\prime \prime \prime }-\left (3 y^{\prime }+a \right ) \left (y^{\prime \prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 743
dsolve((diff(y(x),x)^2+1)*diff(diff(diff(y(x),x),x),x)-(3*diff(y(x),x)+a)*diff(diff(y(x),x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = \int \frac {\sin \left (\RootOf \left (c_{2}^{2} a^{4} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} a^{4} x \,{\mathrm e}^{2 \textit {\_Z} a}+a^{4} x^{2} {\mathrm e}^{2 \textit {\_Z} a}-2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a^{3}-2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a^{3} x +2 c_{2}^{2} a^{2} {\mathrm e}^{2 \textit {\_Z} a}+4 c_{2} a^{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+2 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a -2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a x +c_{2}^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-c_{1}^{2}\right )\right )}{\cos \left (\RootOf \left (c_{2}^{2} a^{4} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} a^{4} x \,{\mathrm e}^{2 \textit {\_Z} a}+a^{4} x^{2} {\mathrm e}^{2 \textit {\_Z} a}-2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a^{3}-2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a^{3} x +2 c_{2}^{2} a^{2} {\mathrm e}^{2 \textit {\_Z} a}+4 c_{2} a^{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+2 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}-2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a -2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a x +c_{2}^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-c_{1}^{2}\right )\right )}d x +c_{3} \\ y \relax (x ) = \int \frac {\sin \left (\RootOf \left (c_{2}^{2} a^{4} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} a^{4} x \,{\mathrm e}^{2 \textit {\_Z} a}+a^{4} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a^{3}+2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a^{3} x +2 c_{2}^{2} a^{2} {\mathrm e}^{2 \textit {\_Z} a}+4 c_{2} a^{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+2 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}+2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a +2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a x +c_{2}^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-c_{1}^{2}\right )\right )}{\cos \left (\RootOf \left (c_{2}^{2} a^{4} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} a^{4} x \,{\mathrm e}^{2 \textit {\_Z} a}+a^{4} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a^{3}+2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a^{3} x +2 c_{2}^{2} a^{2} {\mathrm e}^{2 \textit {\_Z} a}+4 c_{2} a^{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+2 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2} a^{2}+2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} c_{2} a +2 \,{\mathrm e}^{\textit {\_Z} a} \cos \left (\textit {\_Z} \right ) c_{1} a x +c_{2}^{2} {\mathrm e}^{2 \textit {\_Z} a}+2 c_{2} x \,{\mathrm e}^{2 \textit {\_Z} a}+x^{2} {\mathrm e}^{2 \textit {\_Z} a}+\left (\cos ^{2}\left (\textit {\_Z} \right )\right ) c_{1}^{2}-c_{1}^{2}\right )\right )}d x +c_{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.297 (sec). Leaf size: 187
DSolve[(-a - 3*y'[x])*y''[x]^2 + (1 + y'[x]^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_3-\frac {\left (1-i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \text {ArcTan}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right ){}^{-\frac {1}{2}-\frac {i a}{2}} \left (1+i \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \text {ArcTan}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right ){}^{\frac {1}{2} i (a+i)} \left (1+a \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \text {ArcTan}(\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\right )}{\left (a^2+1\right ) c_1} \\ \end{align*}