Internal problem ID [3726]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1003.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {2 x y^{2} \left (y^{\prime }\right )^{2}-y^{3} y^{\prime }-a=0} \end {gather*}
✓ Solution by Maple
Time used: 0.281 (sec). Leaf size: 175
dsolve(2*x*y(x)^2*diff(y(x),x)^2-y(x)^3*diff(y(x),x)-a = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \relax (x ) = -2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \relax (x ) = -i 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \relax (x ) = i 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \relax (x ) = \frac {2^{\frac {1}{4}} \left (a \left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ y \relax (x ) = -\frac {2^{\frac {1}{4}} \left (a \left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ y \relax (x ) = -\frac {i 2^{\frac {1}{4}} \left (a \left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ y \relax (x ) = \frac {i 2^{\frac {1}{4}} \left (a \left (x^{2}-2 c_{1} x +c_{1}^{2}\right ) c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.86 (sec). Leaf size: 220
DSolve[2 x y[x]^2 (y'[x])^2 -y[x]^3 y'[x] -a ==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{-\frac {c_1}{4}} \sqrt {-8 a x+e^{c_1}}}{\sqrt {2}} \\ y(x)\to \frac {e^{-\frac {c_1}{4}} \sqrt {-8 a x+e^{c_1}}}{\sqrt {2}} \\ y(x)\to -\frac {\sqrt {e^{-\frac {c_1}{2}} \left (8 a x-e^{c_1}\right )}}{\sqrt {2}} \\ y(x)\to \frac {\sqrt {e^{-\frac {c_1}{2}} \left (8 a x-e^{c_1}\right )}}{\sqrt {2}} \\ y(x)\to -(-2)^{3/4} \sqrt [4]{a} \sqrt [4]{x} \\ y(x)\to (-2)^{3/4} \sqrt [4]{a} \sqrt [4]{x} \\ y(x)\to (-1-i) \sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x} \\ y(x)\to (1+i) \sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x} \\ \end{align*}