Internal problem ID [3663]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 937.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {3 x^{4} \left (y^{\prime }\right )^{2}-y x -y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.116 (sec). Leaf size: 209
dsolve(3*x^4*diff(y(x),x)^2-x*y(x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = \frac {-\frac {2 \sqrt {3}\, c_{1} x^{2}}{\sqrt {x +1}}-2 \sqrt {3}\, \arctanh \left (\sqrt {x +1}\right ) c_{1} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x^{2}}{\sqrt {x +1}}-\frac {2 \sqrt {3}\, c_{1} x}{\sqrt {x +1}}+\arctanh \left (\sqrt {x +1}\right )^{2} x^{2}+3 c_{1}^{2} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x}{\sqrt {x +1}}+x +1}{12 x^{2}} \\ y \relax (x ) = \frac {\frac {2 \sqrt {3}\, c_{1} x^{2}}{\sqrt {x +1}}+2 \sqrt {3}\, \arctanh \left (\sqrt {x +1}\right ) c_{1} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x^{2}}{\sqrt {x +1}}+\frac {2 \sqrt {3}\, c_{1} x}{\sqrt {x +1}}+\arctanh \left (\sqrt {x +1}\right )^{2} x^{2}+3 c_{1}^{2} x^{2}+\frac {2 \arctanh \left (\sqrt {x +1}\right ) x}{\sqrt {x +1}}+x +1}{12 x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.158 (sec). Leaf size: 165
DSolve[3 x^4 (y'[x])^2-x y[x]-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x+x \left (x \tanh ^{-1}\left (\sqrt {x+1}\right )^2+c_1 \left (-2 \sqrt {3} \sqrt {x+1}+3 c_1 x\right )+2 \left (\sqrt {x+1}-\sqrt {3} c_1 x\right ) \tanh ^{-1}\left (\sqrt {x+1}\right )\right )+1}{12 x^2} \\ y(x)\to \frac {x+x \left (x \tanh ^{-1}\left (\sqrt {x+1}\right )^2+c_1 \left (2 \sqrt {3} \sqrt {x+1}+3 c_1 x\right )+2 \left (\sqrt {x+1}+\sqrt {3} c_1 x\right ) \tanh ^{-1}\left (\sqrt {x+1}\right )\right )+1}{12 x^2} \\ y(x)\to 0 \\ \end{align*}