40.1.3 problem 15

Internal problem ID [6571]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 2. Solutions of differential equations. Supplemetary problems. Page 11
Problem number : 15
Date solved : Monday, January 27, 2025 at 02:12:30 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y&=x y^{\prime }+{y^{\prime }}^{4} \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 64

dsolve(y(x)=x*diff(y(x),x)+diff(y(x),x)^4,y(x), singsol=all)
 

\begin{align*} y &= -\frac {3 \,2^{{1}/{3}} \left (-x \right )^{{4}/{3}}}{8} \\ y &= \frac {3 \,2^{{1}/{3}} \left (-x \right )^{{4}/{3}} \left (1+i \sqrt {3}\right )}{16} \\ y &= -\frac {3 \,2^{{1}/{3}} \left (-x \right )^{{4}/{3}} \left (i \sqrt {3}-1\right )}{16} \\ y &= c_1 \left (c_1^{3}+x \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 75

DSolve[y[x]==x*D[y[x],x]+(D[y[x],x])^4,y[x],x,IncludeSingularSolutions -> True]
 

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