Internal problem ID [8035]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 455.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G]]
Solve \begin {gather*} \boxed {x^{3} \left (y^{\prime }\right )^{2}+x^{2} y y^{\prime }+a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.349 (sec). Leaf size: 48
dsolve(x^3*diff(y(x),x)^2+x^2*y(x)*diff(y(x),x)+a = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c_{1}^{2} x +4 a}{2 c_{1} x} \\ y \relax (x ) = \frac {4 a x +c_{1}^{2}}{2 c_{1} x} \\ y \relax (x ) = \frac {c_{1}}{\sqrt {x}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.989 (sec). Leaf size: 241
DSolve[a + x^2*y[x]*y'[x] + x^3*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {2 y(x) \log \left (\sqrt {x^3 \left (x y(x)^2-4 a\right )}-x^2 \sqrt {y(x)^2}\right )+\left (\sqrt {y(x)^2}-y(x)\right ) \log \left (-y(x) \sqrt {x^3 \left (x y(x)^2-4 a\right )}+2 a x+x^2 y(x) \sqrt {y(x)^2}\right )-\left (y(x)+\sqrt {y(x)^2}\right ) \log (x)}{y(x)}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 y(x) \log \left (\sqrt {x^3 \left (x y(x)^2-4 a\right )}-x^2 \sqrt {y(x)^2}\right )-\left (y(x)+\sqrt {y(x)^2}\right ) \log \left (-y(x) \sqrt {x^3 \left (x y(x)^2-4 a\right )}-2 a x+x^2 y(x) \sqrt {y(x)^2}\right )+\left (\sqrt {y(x)^2}-y(x)\right ) \log (x)}{y(x)}=c_1,y(x)\right ] \\ \end{align*}