87.8.14 problem 14

Internal problem ID [23368]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 65
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:40:42 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {\mathrm e}^{x} {y^{\prime }}^{2}+3 y&=0 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 75

ode:=exp(x)*diff(y(x),x)^2+3*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= \frac {\left (c_1^{2}-12 \,{\mathrm e}^{-x}\right ) \sqrt {-{\mathrm e}^{x}}-4 c_1 \sqrt {3}}{4 \sqrt {-{\mathrm e}^{x}}} \\ y &= \frac {\left (c_1^{2}-12 \,{\mathrm e}^{-x}\right ) \sqrt {-{\mathrm e}^{x}}+4 c_1 \sqrt {3}}{4 \sqrt {-{\mathrm e}^{x}}} \\ \end{align*}

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 82

ode=Exp[x]*D[y[x],{x,1}]^2+3*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -3 e^{-x}-i \sqrt {3} c_1 e^{-x/2}+\frac {c_1{}^2}{4}\\ y(x)&\to -3 e^{-x}+i \sqrt {3} c_1 e^{-x/2}+\frac {c_1{}^2}{4}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.041 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = \frac {C_{1}^{2}}{4} - \sqrt {3} C_{1} \sqrt {- e^{3 x}} e^{- 2 x} - 3 e^{- x} \]