Internal problem ID [3634]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 31
Problem number: 906.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}-3 y y^{\prime } x +x^{3}+2 y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.312 (sec). Leaf size: 41
dsolve(x^2*diff(y(x),x)^2-3*x*diff(y(x),x)*y(x)+x^3+2*y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x \left (c_{1}^{2}+4 x \right )}{2 c_{1}} \\ y \relax (x ) = \frac {x \left (c_{1}^{2} x +4\right )}{2 c_{1}} \\ y \relax (x ) = c_{1} x^{\frac {3}{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.273 (sec). Leaf size: 961
DSolve[x^2 (y'[x])^2-3 x y[x] y'[x]+x^3+2 y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (2 \sqrt [3]{2} e^{3 c_1} x^3+\left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\ y(x)\to \frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}+i\right ) \left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\ y(x)\to -\frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}-i\right ) \left (-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{-4 e^{3 c_1} x^6-e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\ y(x)\to \frac {e^{-\frac {3 c_1}{2}} \left (2 \sqrt [3]{2} e^{3 c_1} x^3+\left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}\right )}{2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\ y(x)\to \frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}+i\right ) \left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}-i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\ y(x)\to -\frac {i e^{-\frac {3 c_1}{2}} \left (\left (\sqrt {3}-i\right ) \left (4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}\right ){}^{2/3}-2 \sqrt [3]{2} \left (\sqrt {3}+i\right ) e^{3 c_1} x^3\right )}{2\ 2^{2/3} \sqrt [3]{4 e^{3 c_1} x^6+e^{6 c_1} x^3+\sqrt {e^{6 c_1} x^6 \left (-4 x^3+e^{3 c_1}\right ){}^2}}} \\ \end{align*}