Internal problem ID [120]
Book: Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section: Section 1.6, Substitution methods and exact equations. Page 74
Problem number: 42.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _exact, _rational]
\[ \boxed {\frac {2 x^{\frac {5}{2}}-3 y^{\frac {5}{3}}}{2 x^{\frac {5}{2}} y^{\frac {2}{3}}}+\frac {\left (-2 x^{\frac {5}{2}}+3 y^{\frac {5}{3}}\right ) y^{\prime }}{3 x^{\frac {3}{2}} y^{\frac {5}{3}}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 189
dsolve(1/2*(2*x^(5/2)-3*y(x)^(5/3))/x^(5/2)/y(x)^(2/3)+1/3*(-2*x^(5/2)+3*y(x)^(5/3))*diff(y(x),x)/x^(3/2)/y(x)^(5/3) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {2^{\frac {3}{5}} 3^{\frac {2}{5}} \left (x^{\frac {5}{2}}\right )^{\frac {3}{5}}}{3} y \left (x \right ) = \frac {\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{3} 2^{\frac {3}{5}} 3^{\frac {2}{5}} \left (x^{\frac {5}{2}}\right )^{\frac {3}{5}}}{3} y \left (x \right ) = \frac {\left (-\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5-\sqrt {5}}}{4}\right )^{3} 2^{\frac {3}{5}} 3^{\frac {2}{5}} \left (x^{\frac {5}{2}}\right )^{\frac {3}{5}}}{3} y \left (x \right ) = \frac {\left (\frac {\sqrt {5}}{4}-\frac {1}{4}-\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{3} 2^{\frac {3}{5}} 3^{\frac {2}{5}} \left (x^{\frac {5}{2}}\right )^{\frac {3}{5}}}{3} y \left (x \right ) = \frac {\left (\frac {\sqrt {5}}{4}-\frac {1}{4}+\frac {i \sqrt {2}\, \sqrt {5+\sqrt {5}}}{4}\right )^{3} 2^{\frac {3}{5}} 3^{\frac {2}{5}} \left (x^{\frac {5}{2}}\right )^{\frac {3}{5}}}{3} \frac {x}{y \left (x \right )^{\frac {2}{3}}}+\frac {y \left (x \right )}{x^{\frac {3}{2}}}+c_{1} = 0 \end{align*}
✓ Solution by Mathematica
Time used: 0.078 (sec). Leaf size: 260
DSolve[1/2*(2*x^(5/2)-3*y[x]^(5/3))/x^(5/2)/y[x]^(2/3)+1/3*(-2*x^(5/2)+3*y[x]^(5/3))*y'[x]/x^(3/2)/y[x]^(5/3) == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (\frac {2}{3}\right )^{3/5} \left (x^{5/2}\right )^{3/5} y(x)\to c_1 x^{3/2} y(x)\to -\left (-\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to \left (-\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to -\left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to -\sqrt [5]{-1} \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to \sqrt [5]{-1} \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to -(-1)^{2/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to (-1)^{2/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to -(-1)^{4/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to (-1)^{4/5} \left (\frac {2}{3}\right )^{3/5} x^{3/2} y(x)\to \left (\frac {2}{3}\right )^{3/5} \left (x^{5/2}\right )^{3/5} \end{align*}