12.6.24 problem 24

Internal problem ID [1703]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number : 24
Date solved : Saturday, March 29, 2025 at 11:35:06 PM
CAS classification : [_exact, _Bernoulli]

\begin{align*} {\mathrm e}^{x} \left (x^{4} y^{2}+4 x^{3} y^{2}+1\right )+\left (2 x^{4} y \,{\mathrm e}^{x}+2 y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 64

ode:=exp(x)*(x^4*y(x)^2+4*x^3*y(x)^2+1)+(2*x^4*y(x)*exp(x)+2*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\left ({\mathrm e}^{x} x^{4}+1\right ) \left (-{\mathrm e}^{x}+c_1 \right )}}{{\mathrm e}^{x} x^{4}+1} \\ y &= -\frac {\sqrt {\left ({\mathrm e}^{x} x^{4}+1\right ) \left (-{\mathrm e}^{x}+c_1 \right )}}{{\mathrm e}^{x} x^{4}+1} \\ \end{align*}

✓ Mathematica. Time used: 1.146 (sec). Leaf size: 64

ode=(Exp[x]*(x^4*y[x]^2+4*x^3*y[x]^2+1))+(2*x^4*y[x]*Exp[x]+2*y[x])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.939 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x**4*y(x)*exp(x) + 2*y(x))*Derivative(y(x), x) + (x**4*y(x)**2 + 4*x**3*y(x)**2 + 1)*exp(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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