15.21.5 problem 27

Internal problem ID [3329]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 39, page 179
Problem number : 27
Date solved : Monday, January 27, 2025 at 07:34:18 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 70

dsolve(y(x)=diff(y(x),x)*x+3/diff(y(x),x)^2,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {3 \,6^{{1}/{3}} \left (x^{2}\right )^{{1}/{3}}}{2} \\ y \left (x \right ) &= -\frac {3 \,2^{{1}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) \left (x^{2}\right )^{{1}/{3}}}{4} \\ y \left (x \right ) &= -\frac {3 \,2^{{1}/{3}} \left (-i 3^{{5}/{6}}+3^{{1}/{3}}\right ) \left (x^{2}\right )^{{1}/{3}}}{4} \\ y \left (x \right ) &= c_{1} x +\frac {3}{c_{1}^{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 84

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

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