48.3.2 problem Example 3.30

Internal problem ID [7526]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.5 HIGHER ORDER ODE. Page 181
Problem number : Example 3.30
Date solved : Monday, January 27, 2025 at 03:04:36 PM
CAS classification : [[_3rd_order, _missing_x], [_3rd_order, _missing_y], [_3rd_order, _with_exponential_symmetries], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

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

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 38

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

\begin{align*} y &= c_{1} \\ y &= \frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {c_3 +x}{c_{1}}}}{c_{2} c_{1}}\right ) c_{1} -c_3 -x}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.035 (sec). Leaf size: 79

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

\[ y(x)\to \log \left (\text {InverseFunction}\left [-\frac {1}{\text {$\#$1}}-c_1 \log (\text {$\#$1})+c_1 \log (1+\text {$\#$1} c_1)\&\right ][x+c_2]\right )-\log \left (1+c_1 \text {InverseFunction}\left [-\frac {1}{\text {$\#$1}}-c_1 \log (\text {$\#$1})+c_1 \log (1+\text {$\#$1} c_1)\&\right ][x+c_2]\right )+c_3 \]