60.8.11 problem 1847

Internal problem ID [11846]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1847
Date solved : Monday, January 27, 2025 at 11:43:33 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

\begin{align*} \left ({y^{\prime }}^{2}+1\right ) y^{\prime \prime \prime }-3 y^{\prime } {y^{\prime \prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.216 (sec). Leaf size: 65

dsolve((diff(y(x),x)^2+1)*diff(diff(diff(y(x),x),x),x)-3*diff(y(x),x)*diff(diff(y(x),x),x)^2=0,y(x), singsol=all)
 

\begin{align*} y &= -i x +c_{1} \\ y &= i x +c_{1} \\ y &= -\sqrt {-c_{2}^{2}-2 c_{2} x -x^{2}+c_{1}}+c_3 \\ y &= \sqrt {-c_{2}^{2}-2 c_{2} x -x^{2}+c_{1}}+c_3 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.133 (sec). Leaf size: 142

DSolve[-3*D[y[x],x]*D[y[x],{x,2}]^2 + (1 + D[y[x],x]^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_3-\frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1} \\ y(x)\to \frac {i \sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}}{c_1}+c_3 \\ y(x)\to \text {Indeterminate} \\ y(x)\to c_3-i \sqrt {(x+c_2){}^2} \\ y(x)\to i \sqrt {(x+c_2){}^2}+c_3 \\ \end{align*}