Internal problem ID [4237]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1012.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {3 x y^{4} {y^{\prime }}^{2}-y^{5} y^{\prime }=-1} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 297
dsolve(3*x*y(x)^4*diff(y(x),x)^2-y(x)^5*diff(y(x),x)+1 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 2^{\frac {1}{3}} 3^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \left (x \right ) &= -2^{\frac {1}{3}} 3^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) 3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.332 (sec). Leaf size: 230
DSolve[3 x y[x]^4 (y'[x])^2 -y[x]^5 y'[x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt [3]{-\frac {1}{2}} e^{-\frac {c_1}{6}} \sqrt [3]{12 x+e^{c_1}} \\ y(x)\to e^{-\frac {c_1}{6}} \sqrt [3]{6 x+\frac {e^{c_1}}{2}} \\ y(x)\to (-1)^{2/3} e^{-\frac {c_1}{6}} \sqrt [3]{6 x+\frac {e^{c_1}}{2}} \\ y(x)\to -\sqrt [3]{-2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to \sqrt [3]{-2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to -\sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to -(-1)^{2/3} \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\ y(x)\to (-1)^{2/3} \sqrt [3]{2} \sqrt [6]{3} \sqrt [6]{x} \\ \end{align*}