29.33.29 problem 992

Internal problem ID [5568]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 992
Date solved : Monday, January 27, 2025 at 12:06:38 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

Solution by Maple

Time used: 0.107 (sec). Leaf size: 105

dsolve(3*y(x)^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)-x^2+4*y(x)^2 = 0,y(x), singsol=all)
 
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, x}{3} \\ y \left (x \right ) &= \frac {\sqrt {3}\, x}{3} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \end{align*}

Solution by Mathematica

Time used: 0.574 (sec). Leaf size: 179

DSolve[3  y[x]^2 (D[y[x],x])^2 -2 x y[x] D[y[x],x]-x^2+4 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 
\begin{align*} y(x)\to -\frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to \frac {\sqrt {-3 x^2-4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to -\frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to \frac {\sqrt {-3 x^2+4 i e^{3 c_1} x+e^{6 c_1}}}{\sqrt {3}} \\ y(x)\to -\sqrt {-x^2} \\ y(x)\to \sqrt {-x^2} \\ \end{align*}