ODE No. 669

$$y^{\prime }(x)=\frac{e^x{(3e^x-2y(x{)}^{3/2})}^2}{4\sqrt{y(x)}}$$ ✓ Mathematica : cpu = 0.712139 (sec), leaf count = 264

DSolve[Derivative[1][y][x] == (E^x*(3*E^x - 2*y[x]^(3/2))^2)/(4*Sqrt[y[x]]),y[x],x]
 

$$\{\{y(x)\to \frac{(-2e^{3e^x}+3e^{x+3e^x}+3e^{x+3c_1}+2e^{3c_1}){}^{2/3}}{\sqrt[3]{4e^{6e^x}+8e^{3e^x+3c_1}+4e^{6c_1}}}\},\{y(x)\to -\frac{\sqrt[3]{-1}(-2e^{3e^x}+3e^{x+3e^x}+3e^{x+3c_1}+2e^{3c_1}){}^{2/3}}{\sqrt[3]{4e^{6e^x}+8e^{3e^x+3c_1}+4e^{6c_1}}}\},\{y(x)\to \frac{(-1{)}^{2/3}(-2e^{3e^x}+3e^{x+3e^x}+3e^{x+3c_1}+2e^{3c_1}){}^{2/3}}{\sqrt[3]{4e^{6e^x}+8e^{3e^x+3c_1}+4e^{6c_1}}}\}\}$$ ✓ Maple : cpu = 0.19 (sec), leaf count = 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))
 

$$c_1+\frac{{\mathrm{e}}^{-\frac{3{\mathrm{e}}^x}{2}-\frac{9{\mathrm{e}}^{2x}}{8}}(2y{\left(x\right)}^{\frac{3}{2}}{\mathrm{e}}^x-2{\mathrm{e}}^x-3{\mathrm{e}}^{2x}){\mathrm{e}}^{-\frac{3{\mathrm{e}}^x}{2}+\frac{9{\mathrm{e}}^{2x}}{8}}}{2y{\left(x\right)}^{\frac{3}{2}}{\mathrm{e}}^x+2{\mathrm{e}}^x-3{\mathrm{e}}^{2x}}=0$$