28.1.82 problem 85

Internal problem ID [4388]
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 : 85
Date solved : Monday, January 27, 2025 at 09:09:50 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} 2 {y^{\prime }}^{2} \left (y-x y^{\prime }\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 51

dsolve(2*(diff(y(x),x))^2*(y(x)-x*diff(y(x),x))=1,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 67

DSolve[2*(D[y[x],x])^2*(y[x]-x*D[y[x],x])==1,y[x],x,IncludeSingularSolutions -> True]
 

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