Internal problem ID [3556]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 825.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}-2 x^{3} y^{2} y^{\prime }-4 y^{3} x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.195 (sec). Leaf size: 135
dsolve(diff(y(x),x)^2-2*x^3*y(x)^2*diff(y(x),x)-4*x^2*y(x)^3 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {4}{x^{4}} \\ y \relax (x ) = 0 \\ y \relax (x ) = \frac {\left (\sqrt {2}\, x^{2} c_{1}-2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{4}-4} \\ y \relax (x ) = -\frac {\left (\sqrt {2}\, x^{2} c_{1}+2\right ) c_{1}^{2}}{2 \left (c_{1}^{2} x^{4}-2\right )} \\ y \relax (x ) = -\frac {2 \left (\sqrt {2}\, x^{2} c_{1}-c_{1}^{2}\right )}{c_{1}^{2} \left (-2 x^{4}+c_{1}^{2}\right )} \\ y \relax (x ) = \frac {2 \sqrt {2}\, x^{2} c_{1}+2 c_{1}^{2}}{c_{1}^{2} \left (-2 x^{4}+c_{1}^{2}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.282 (sec). Leaf size: 175
DSolve[(y'[x])^2-2 x^3 y[x]^2 y'[x]-4 x^2 y[x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+4} y(x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x^4 y(x)+4}}{x^2 \sqrt {y(x)}}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x y(x)^{3/2} \sqrt {x^4 y(x)+4} \tanh ^{-1}\left (\frac {\sqrt {x^4 y(x)+4}}{x^2 \sqrt {y(x)}}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to -\frac {4}{x^4} \\ \end{align*}