20.5.10 problem Problem 10

Internal problem ID [3693]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.9, Exact Differential Equations. page 91
Problem number : Problem 10
Date solved : Sunday, March 30, 2025 at 02:05:53 AM
CAS classification : [_exact, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 7.876 (sec). Leaf size: 47

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

\begin{align*} y(x)\to -\sqrt {e^{-2 x} \left (-x^3+c_1\right )} \\ y(x)\to \sqrt {e^{-2 x} \left (-x^3+c_1\right )} \\ \end{align*}

✓ Sympy. Time used: 0.604 (sec). Leaf size: 32

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

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