60.2.93 problem 669

Internal problem ID [10680]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 669
Date solved : Tuesday, January 28, 2025 at 05:04:46 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {\left (-2 y^{{3}/{2}}+3 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{x}}{4 \sqrt {y}} \end{align*}

✓ Solution by Maple

Time used: 0.006 (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)
 

\[ \frac {\left (3 \,{\mathrm e}^{2 x -\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+3 c_{1} {\mathrm e}^{2 x +\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+2 \left (1-y^{{3}/{2}}\right ) {\mathrm e}^{x -\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}+2 c_{1} \left (-1-y^{{3}/{2}}\right ) {\mathrm e}^{x +\frac {3 \,{\mathrm e}^{x}}{2}-\frac {9 \,{\mathrm e}^{2 x}}{8}}\right ) {\mathrm e}^{-x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {9 \,{\mathrm e}^{2 x}}{8}}}{-2 y^{{3}/{2}}+3 \,{\mathrm e}^{x}-2} = 0 \]

✓ Solution by Mathematica

Time used: 60.766 (sec). Leaf size: 222

DSolve[D[y[x],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*}