40.5.11 problem 27

Internal problem ID [6676]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 27
Date solved : Monday, January 27, 2025 at 02:19:02 PM
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: 0.086 (sec). Leaf size: 95

dsolve(y(x)=2*x*diff(y(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: 0.125 (sec). Leaf size: 119

DSolve[y[x]==2*x*D[y[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*}