Internal problem ID [4171]
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)]`]]
\[ \boxed {3 x^{4} {y^{\prime }}^{2}-y x -y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 223
dsolve(3*x^4*diff(y(x),x)^2-x*y(x)-y(x) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = 0 y \left (x \right ) = \frac {\left (-\operatorname {arctanh}\left (\frac {x}{\sqrt {x +1}}+\frac {1}{\sqrt {x +1}}\right ) \sqrt {3}\, x -\frac {\sqrt {3}\, x}{\sqrt {x +1}}-\frac {\sqrt {3}}{\sqrt {x +1}}-3 c_{1} x \right )^{2}}{36 x^{2}} y \left (x \right ) = \frac {\left (\operatorname {arctanh}\left (\frac {x}{\sqrt {x +1}}+\frac {1}{\sqrt {x +1}}\right ) \sqrt {3}\, x +\frac {\sqrt {3}\, x}{\sqrt {x +1}}+\frac {\sqrt {3}}{\sqrt {x +1}}-3 c_{1} x \right )^{2}}{36 x^{2}} y \left (x \right ) = \frac {\left (-\operatorname {arctanh}\left (\frac {x}{\sqrt {x +1}}+\frac {1}{\sqrt {x +1}}\right ) \sqrt {3}\, x -\frac {\sqrt {3}\, x}{\sqrt {x +1}}-\frac {\sqrt {3}}{\sqrt {x +1}}+3 c_{1} x \right )^{2}}{36 x^{2}} y \left (x \right ) = \frac {\left (\operatorname {arctanh}\left (\frac {x}{\sqrt {x +1}}+\frac {1}{\sqrt {x +1}}\right ) \sqrt {3}\, x +\frac {\sqrt {3}\, x}{\sqrt {x +1}}+\frac {\sqrt {3}}{\sqrt {x +1}}+3 c_{1} x \right )^{2}}{36 x^{2}} \end{align*}
✓ Solution by Mathematica
Time used: 0.153 (sec). Leaf size: 171
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^2 \text {arctanh}\left (\sqrt {x+1}\right )^2+2 x \text {arctanh}\left (\sqrt {x+1}\right ) \left (\sqrt {x+1}-\sqrt {3} c_1 x\right )+3 c_1{}^2 x^2+x-2 \sqrt {3} c_1 x \sqrt {x+1}+1}{12 x^2} y(x)\to \frac {x^2 \text {arctanh}\left (\sqrt {x+1}\right )^2+2 x \text {arctanh}\left (\sqrt {x+1}\right ) \left (\sqrt {x+1}+\sqrt {3} c_1 x\right )+3 c_1{}^2 x^2+x+2 \sqrt {3} c_1 x \sqrt {x+1}+1}{12 x^2} y(x)\to 0 \end{align*}