Internal problem ID [8249]
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)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (-2 y^{\frac {3}{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 72
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)
\[ c_{1}+\frac {{\mathrm e}^{-\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}} \left (2 y \relax (x )^{\frac {3}{2}} {\mathrm e}^{x}-2 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-\frac {3 \,{\mathrm e}^{x}}{2}+\frac {9 \,{\mathrm e}^{2 x}}{8}}}{2 y \relax (x )^{\frac {3}{2}} {\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}} = 0 \]
✓ Solution by Mathematica
Time used: 60.791 (sec). Leaf size: 201
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} \left (3 e^x-2\right )+e^{3 c_1} \left (3 e^x+2\right )\right ){}^{2/3}}{2^{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} \left (3 e^x-2\right )+e^{3 c_1} \left (3 e^x+2\right )\right ){}^{2/3}}{2^{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 (e^{3 e^x} \left (3 e^x-2\right )+e^{3 c_1} \left (3 e^x+2\right )\right ){}^{2/3}}{\sqrt [3]{\left (e^{3 e^x}+e^{3 c_1}\right ){}^2}} \\ \end{align*}