Internal problem ID [4064]
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]]
\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } y^{2} x^{3}-4 y^{3} x^{2}=0} \]
✓ Solution by Maple
Time used: 0.157 (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 \left (x \right ) = -\frac {4}{x^{4}} y \left (x \right ) = 0 y \left (x \right ) = \frac {\left (\sqrt {2}\, x^{2} c_{1} -2\right ) c_{1}^{2}}{2 c_{1}^{2} x^{4}-4} y \left (x \right ) = -\frac {\left (\sqrt {2}\, x^{2} c_{1} +2\right ) c_{1}^{2}}{2 \left (c_{1}^{2} x^{4}-2\right )} y \left (x \right ) = -\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 \left (x \right ) = \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: 1.481 (sec). Leaf size: 177
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} \log \left (\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} \log \left (\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*}