15.19.20 problem 20

Internal problem ID [3304]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 20
Date solved : Monday, January 27, 2025 at 07:32:49 AM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

\begin{align*} 8 x +1&=y {y^{\prime }}^{2} \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 137

dsolve(8*x+1=diff(y(x),x)^2*y(x),y(x), singsol=all)
 

\begin{align*} -\frac {8 c_{1} \left (8 x +1\right )}{\left (\frac {-\sqrt {y \left (x \right ) \left (8 x +1\right )}-2 y \left (x \right )}{y \left (x \right )}\right )^{{2}/{3}} y \left (x \right ) \left (\frac {8 x +1-2 \sqrt {y \left (x \right ) \left (8 x +1\right )}+4 y \left (x \right )}{y \left (x \right )}\right )^{{2}/{3}}}+x +\frac {1}{8} &= 0 \\ -\frac {8 c_{1} \left (8 x +1\right )}{\left (\frac {\sqrt {y \left (x \right ) \left (8 x +1\right )}-2 y \left (x \right )}{y \left (x \right )}\right )^{{2}/{3}} y \left (x \right ) \left (\frac {8 x +1+2 \sqrt {y \left (x \right ) \left (8 x +1\right )}+4 y \left (x \right )}{y \left (x \right )}\right )^{{2}/{3}}}+x +\frac {1}{8} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.706 (sec). Leaf size: 79

DSolve[8*x+1==D[y[x],x]^2*y[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {1}{4} \left (-8 \sqrt {8 x+1} x-\sqrt {8 x+1}+12 c_1\right ){}^{2/3} \\ y(x)\to \frac {1}{4} \left (8 \sqrt {8 x+1} x+\sqrt {8 x+1}+12 c_1\right ){}^{2/3} \\ \end{align*}