Internal problem ID [4234]
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]
\[ \boxed {2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }=a} \]
✓ Solution by Maple
Time used: 0.125 (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 \left (x \right ) = 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} y \left (x \right ) = -2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} y \left (x \right ) = -i 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} y \left (x \right ) = i 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} y \left (x \right ) = \frac {2^{\frac {1}{4}} {\left (a \left (c_{1}^{2}-2 c_{1} x +x^{2}\right ) c_{1}^{3}\right )}^{\frac {1}{4}}}{c_{1}} y \left (x \right ) = -\frac {2^{\frac {1}{4}} {\left (a \left (c_{1}^{2}-2 c_{1} x +x^{2}\right ) c_{1}^{3}\right )}^{\frac {1}{4}}}{c_{1}} y \left (x \right ) = -\frac {i 2^{\frac {1}{4}} {\left (a \left (c_{1}^{2}-2 c_{1} x +x^{2}\right ) c_{1}^{3}\right )}^{\frac {1}{4}}}{c_{1}} y \left (x \right ) = \frac {i 2^{\frac {1}{4}} {\left (a \left (c_{1}^{2}-2 c_{1} x +x^{2}\right ) c_{1}^{3}\right )}^{\frac {1}{4}}}{c_{1}} \end{align*}
✓ Solution by Mathematica
Time used: 1.666 (sec). Leaf size: 151
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 -(-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*}