60.1.530 problem 533

Internal problem ID [10544]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 533
Date solved : Monday, January 27, 2025 at 08:53:34 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 74

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

\begin{align*} y &= \frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}}}{2} \\ y &= -\frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4} \\ y &= \frac {c_{1}^{3} x +a}{c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 89

DSolve[a - y[x]*D[y[x],x]^2 + x*D[y[x],x]^3==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {a}{c_1{}^2}+c_1 x \\ y(x)\to \frac {3 \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ y(x)\to -\frac {3 \sqrt [3]{-1} \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ y(x)\to \frac {3 (-1)^{2/3} \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ \end{align*}