Internal problem ID [4081]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 29
Problem number: 842.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
\[ \boxed {9 {y^{\prime }}^{2}+3 x y^{4} y^{\prime }+y^{5}=0} \]
✓ Solution by Maple
Time used: 0.265 (sec). Leaf size: 102
dsolve(9*diff(y(x),x)^2+3*x*y(x)^4*diff(y(x),x)+y(x)^5 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}}}{x^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{2 x^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{2 x^{\frac {2}{3}}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}+\sqrt {\textit {\_a}^{3} \left (\textit {\_a}^{3}-4\right )}-4}{\textit {\_a} \left (\textit {\_a}^{3}-4\right )}d \textit {\_a} \right )+2 c_{1} \right )}{x^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.075 (sec). Leaf size: 212
DSolve[9 (y'[x])^2+3 x y[x]^4 y'[x]+y[x]^5==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {\sqrt {x^2 y(x)^3-4} y(x)^{5/2} \text {arctanh}\left (\frac {x y(x)^{3/2}}{\sqrt {x^2 y(x)^3-4}}\right )}{\sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}-\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {y(x)^{5/2} \sqrt {x^2 y(x)^3-4} \text {arctanh}\left (\frac {x y(x)^{3/2}}{\sqrt {x^2 y(x)^3-4}}\right )}{\sqrt {y(x)^5 \left (x^2 y(x)^3-4\right )}}-\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to \frac {(-2)^{2/3}}{x^{2/3}} \\ y(x)\to \frac {2^{2/3}}{x^{2/3}} \\ y(x)\to -\frac {\sqrt [3]{-1} 2^{2/3}}{x^{2/3}} \\ \end{align*}