62.16.7 problem Ex 7

Internal problem ID [12898]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 27. Clairaut equation. Page 56
Problem number : Ex 7
Date solved : Tuesday, January 28, 2025 at 04:37:37 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.138 (sec). Leaf size: 95

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

\begin{align*} y &= -\frac {2 \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= \frac {2 \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= -\frac {2 i \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= \frac {2 i \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y &= 0 \\ y &= \sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ y &= -\sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.130 (sec). Leaf size: 119

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

\begin{align*} y(x)\to -\sqrt {2 c_1 x+c_1{}^3} \\ y(x)\to \sqrt {2 c_1 x+c_1{}^3} \\ y(x)\to (-1-i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (1-i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (-1+i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ y(x)\to (1+i) \left (\frac {2}{3}\right )^{3/4} x^{3/4} \\ \end{align*}