49.23.3 problem 1(c)

Internal problem ID [7761]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page 238
Problem number : 1(c)
Date solved : Monday, January 27, 2025 at 03:12:12 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+4 {y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 152

dsolve(y(x)*diff(y(x),x$2)+4*diff(y(x),x)^2=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \left (5 c_{1} x +5 c_{2} \right )^{{1}/{5}} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left (5 c_{1} x +5 c_{2} \right )^{{1}/{5}}}{4} \\ y &= -\frac {\left (-i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left (5 c_{1} x +5 c_{2} \right )^{{1}/{5}}}{4} \\ y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) \left (5 c_{1} x +5 c_{2} \right )^{{1}/{5}}}{4} \\ y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left (5 c_{1} x +5 c_{2} \right )^{{1}/{5}}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.176 (sec). Leaf size: 20

DSolve[y[x]*D[y[x],{x,2}]+4*(D[y[x],x])^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \sqrt [5]{5 x-c_1} \]