69.1.83 problem 128

Internal problem ID [14236]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 128
Date solved : Tuesday, January 28, 2025 at 06:22:58 AM
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*} y^{\prime } y^{\prime \prime \prime }-3 {y^{\prime \prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.216 (sec). Leaf size: 59

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

\begin{align*} y &= c_{1} \\ y &= \frac {-c_{2} c_{1} +\sqrt {-2 \left (-\frac {c_{1} c_{2}^{2}}{2}+x +c_{3} \right ) c_{1}}}{c_{1}} \\ y &= \frac {-c_{2} c_{1} -\sqrt {-2 \left (-\frac {c_{1} c_{2}^{2}}{2}+x +c_{3} \right ) c_{1}}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.185 (sec). Leaf size: 21

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

\[ y(x)\to c_2 \sqrt {2 x+c_1}+c_3 \]