Internal problem ID [5340]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y y^{\prime \prime }+4 \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.09 (sec). Leaf size: 158
dsolve(y(x)*diff(y(x),x$2)+4*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \left (5 c_{1} x +5 c_{2}\right )^{\frac {1}{5}} \\ y \relax (x ) = \left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} x +5 c_{2}\right )^{\frac {1}{5}} \\ y \relax (x ) = \left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right ) \left (5 c_{1} x +5 c_{2}\right )^{\frac {1}{5}} \\ y \relax (x ) = \left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} x +5 c_{2}\right )^{\frac {1}{5}} \\ y \relax (x ) = \left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right ) \left (5 c_{1} x +5 c_{2}\right )^{\frac {1}{5}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.072 (sec). Leaf size: 20
DSolve[y[x]*y''[x]+4*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \sqrt [5]{5 x-c_1} \\ \end{align*}