Internal problem ID [9004]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 669.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\[ \boxed {y^{\prime }-\frac {\left (-2 y^{\frac {3}{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 116
dsolve(diff(y(x),x) = 1/4*(-2*y(x)^(3/2)+3*exp(x))^2*exp(x)/y(x)^(1/2),y(x), singsol=all)
\[ \frac {\left (3 \,{\mathrm e}^{2 x -\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+3 c_{1} {\mathrm e}^{2 x +\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+2 \left (1-y \left (x \right )^{\frac {3}{2}}\right ) {\mathrm e}^{x -\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+2 c_{1} \left (-1-y \left (x \right )^{\frac {3}{2}}\right ) {\mathrm e}^{x +\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}\right ) {\mathrm e}^{-x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {9 \,{\mathrm e}^{2 x}}{8}}}{-2 y \left (x \right )^{\frac {3}{2}}+3 \,{\mathrm e}^{x}-2} = 0 \]
✓ Solution by Mathematica
Time used: 60.755 (sec). Leaf size: 222
DSolve[y'[x] == (E^x*(3*E^x - 2*y[x]^(3/2))^2)/(4*Sqrt[y[x]]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\left (-e^{3 e^x}+\frac {3}{2} e^{x+3 e^x}+\frac {3}{2} e^{x+3 c_1}+e^{3 c_1}\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ y(x)\to -\frac {\sqrt [3]{-1} \left (-e^{3 e^x}+\frac {3}{2} e^{x+3 e^x}+\frac {3}{2} e^{x+3 c_1}+e^{3 c_1}\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ y(x)\to \frac {\left (-\frac {1}{2}\right )^{2/3} \left (-2 e^{3 e^x}+3 e^{x+3 e^x}+3 e^{x+3 c_1}+2 e^{3 c_1}\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ \end{align*}