62.7.4 problem Ex 4

Internal problem ID [12756]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 14. Equations reducible to linear equations (Bernoulli). Page 21
Problem number : Ex 4
Date solved : Monday, March 31, 2025 at 07:02:50 AM
CAS classification : [_Bernoulli]

\begin{align*} 4 x y^{\prime }+3 y+{\mathrm e}^{x} x^{4} y^{5}&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 67

ode:=4*x*diff(y(x),x)+3*y(x)+exp(x)*x^4*y(x)^5 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 15.152 (sec). Leaf size: 88

ode=4*x*D[y[x],x]+3*y[x]+Exp[x]*x^4*y[x]^5==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt [4]{x^3 \left (e^x+c_1\right )}} \\ y(x)\to -\frac {i}{\sqrt [4]{x^3 \left (e^x+c_1\right )}} \\ y(x)\to \frac {i}{\sqrt [4]{x^3 \left (e^x+c_1\right )}} \\ y(x)\to \frac {1}{\sqrt [4]{x^3 \left (e^x+c_1\right )}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 3.393 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*y(x)**5*exp(x) + 4*x*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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