29.30.28 problem 888

Internal problem ID [5468]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 30
Problem number : 888
Date solved : Monday, January 27, 2025 at 11:24:53 AM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 86

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

\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {-x}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {-x}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{x} \\ y \left (x \right ) &= -\frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.524 (sec). Leaf size: 80

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

\begin{align*} y(x)\to -\frac {2 e^{\frac {c_1}{2}}}{-x+e^{c_1}} \\ y(x)\to \frac {2 e^{\frac {c_1}{2}}}{-x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i}{\sqrt {x}} \\ y(x)\to \frac {i}{\sqrt {x}} \\ \end{align*}