60.8.9 problem 1845

Internal problem ID [11844]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 7, non-linear third and higher order
Problem number : 1845
Date solved : Tuesday, January 28, 2025 at 06:23:54 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _with_linear_symmetries]]

\begin{align*} 9 y^{2} y^{\prime \prime \prime }-45 y y^{\prime } y^{\prime \prime }+40 {y^{\prime }}^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.140 (sec). Leaf size: 85

dsolve(9*y(x)^2*diff(diff(diff(y(x),x),x),x)-45*y(x)*diff(y(x),x)*diff(diff(y(x),x),x)+40*diff(y(x),x)^3=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (-6 \left (\int _{}^{\textit {\_Z}}\frac {1}{4 \textit {\_h}^{2}+\sqrt {c_{1} \left (4 \textit {\_h}^{2}+c_{1} \right )}+c_{1}}d \textit {\_h} \right )+x +c_{2} \right )d x +c_3} \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (6 \left (\int _{}^{\textit {\_Z}}-\frac {1}{4 \textit {\_h}^{2}-\sqrt {c_{1} \left (4 \textit {\_h}^{2}+c_{1} \right )}+c_{1}}d \textit {\_h} \right )+x +c_{2} \right )d x +c_3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.110 (sec). Leaf size: 21

DSolve[40*D[y[x],x]^3 - 45*y[x]*D[y[x],x]*D[y[x],{x,2}] + 9*y[x]^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{(x (c_3 x+c_2)+c_1){}^{3/2}} \]