28.1.83 problem 86

Internal problem ID [4389]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 86
Date solved : Monday, January 27, 2025 at 09:09:51 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: 0.115 (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 \left (x \right ) &= -\frac {2 \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y \left (x \right ) &= \frac {2 \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y \left (x \right ) &= -\frac {2 i \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y \left (x \right ) &= \frac {2 i \left (-x^{3}\right )^{{1}/{4}} 6^{{1}/{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.119 (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*}