29.34.9 problem 1006

Internal problem ID [5580]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 1006
Date solved : Monday, January 27, 2025 at 12:11:31 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} 4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.207 (sec). Leaf size: 82

dsolve(4*y(x)^3*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \sqrt {-x} \\ y \left (x \right ) &= -\sqrt {-x} \\ y \left (x \right ) &= \sqrt {x} \\ y \left (x \right ) &= -\sqrt {x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{4}+\sqrt {-\textit {\_a}^{4}+1}-1}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_{1} \right ) \sqrt {x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.332 (sec). Leaf size: 150

DSolve[4 y[x]^3 (D[y[x],x])^2 -4 x D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to -i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to i \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to \sqrt [4]{-e^{c_1} \left (-2 x+e^{c_1}\right )} \\ y(x)\to 0 \\ y(x)\to -\sqrt {x} \\ y(x)\to -i \sqrt {x} \\ y(x)\to i \sqrt {x} \\ y(x)\to \sqrt {x} \\ \end{align*}