69.6.28 problem 161

Internal problem ID [18071]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 161
Date solved : Sunday, October 12, 2025 at 05:33:46 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{\prime }-2 y \,{\mathrm e}^{x}&=2 \sqrt {y \,{\mathrm e}^{x}} \end{align*}

✓ Maple. Time used: 0.532 (sec). Leaf size: 37

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

\[ c_1 +\frac {y \,{\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}}}{\sqrt {y \,{\mathrm e}^{x}}}-\int {\mathrm e}^{\frac {x}{2}-{\mathrm e}^{x}}d x = 0 \]

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 56

ode=D[y[x],x]-2*y[x]*Exp[x]==2*Sqrt[y[x]*Exp[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {2 \left (\sqrt {\pi } \sqrt {y(x)} \text {erf}\left (\frac {\sqrt {e^x y(x)}}{\sqrt {y(x)}}\right )-e^{-e^x} y(x)\right )}{\sqrt {y(x)}}=c_1,y(x)\right ] \]

✗ Sympy

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

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