54.2.93 problem 669

Internal problem ID [11967]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 669
Date solved : Sunday, October 12, 2025 at 02:07:20 AM
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*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 65

ode:=diff(y(x),x) = 1/4*(-2*y(x)^(3/2)+3*exp(x))^2*exp(x)/y(x)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \frac {\left (2 y^{{3}/{2}} c_1 \,{\mathrm e}^{3 \,{\mathrm e}^{x}}-3 c_1 \,{\mathrm e}^{x +3 \,{\mathrm e}^{x}}+2 c_1 \,{\mathrm e}^{3 \,{\mathrm e}^{x}}+2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}-2\right ) {\mathrm e}^{-3 \,{\mathrm e}^{x}}}{2 y^{{3}/{2}}-3 \,{\mathrm e}^{x}+2} = 0 \]

✓ Mathematica. Time used: 60.875 (sec). Leaf size: 222

ode=D[y[x],x] == (E^x*(3*E^x - 2*y[x]^(3/2))^2)/(4*Sqrt[y[x]]); 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(-2*y(x)**(3/2) + 3*exp(x))**2*exp(x)/(4*sqrt(y(x))) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -(-3*y(x)**(3/2)*exp(x) + y(x)**3 + 9*exp(2*x)/4)*exp(x)/sqrt(y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method