Internal problem ID [3716]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 33
Problem number: 992.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {3 y^{2} \left (y^{\prime }\right )^{2}-2 y y^{\prime } x -x^{2}+4 y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 203
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 \relax (x ) = -\frac {\sqrt {3}\, x}{3} \\ y \relax (x ) = \frac {\sqrt {3}\, x}{3} \\ \ln \relax (x )-\frac {\sqrt {3}\, \sqrt {\frac {\left (\sqrt {3}\, x +3 y \relax (x )\right ) \left (\sqrt {3}\, x -3 y \relax (x )\right )}{x^{2}}}}{6}+\frac {\sqrt {\frac {x^{2}-3 y \relax (x )^{2}}{x^{2}}}}{2}-\arctanh \left (\frac {\sqrt {\frac {x^{2}-3 y \relax (x )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \relax (x )^{2}}{x^{2}}\right )}{2}-c_{1} = 0 \\ \ln \relax (x )+\frac {\sqrt {3}\, \sqrt {\frac {\left (\sqrt {3}\, x +3 y \relax (x )\right ) \left (\sqrt {3}\, x -3 y \relax (x )\right )}{x^{2}}}}{6}-\frac {\sqrt {\frac {x^{2}-3 y \relax (x )^{2}}{x^{2}}}}{2}+\arctanh \left (\frac {\sqrt {\frac {x^{2}-3 y \relax (x )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \relax (x )^{2}}{x^{2}}\right )}{2}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.578 (sec). Leaf size: 179
DSolve[3 y[x]^2 (y'[x])^2 -2 x y[x] y'[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*}