60.2.17 problem 593

Internal problem ID [10591]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 593
Date solved : Sunday, March 30, 2025 at 06:09:03 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 35

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

\[ \int _{\textit {\_b}}^{y}\frac {\sqrt {\textit {\_a}}}{F \left (\textit {\_a}^{{3}/{2}}-\frac {3 \,{\mathrm e}^{x}}{2}\right )-1}d \textit {\_a} -{\mathrm e}^{x}-c_1 = 0 \]

✓ Mathematica. Time used: 0.236 (sec). Leaf size: 221

ode=D[y[x],x] == (E^x*F[(-3*E^x)/2 + y[x]^(3/2)])/Sqrt[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {\sqrt {K[2]}}{F\left (K[2]^{3/2}-\frac {3 e^x}{2}\right )-1}-\int _1^x\left (\frac {3 e^{K[1]} F\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right ) \sqrt {K[2]} F''\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )}{2 \left (F\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )-1\right )^2}-\frac {3 e^{K[1]} \sqrt {K[2]} F''\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )}{2 \left (F\left (K[2]^{3/2}-\frac {3 e^{K[1]}}{2}\right )-1\right )}\right )dK[1]\right )dK[2]+\int _1^x-\frac {e^{K[1]} F\left (y(x)^{3/2}-\frac {3 e^{K[1]}}{2}\right )}{F\left (y(x)^{3/2}-\frac {3 e^{K[1]}}{2}\right )-1}dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

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