56.12.5 problem Ex 5

Internal problem ID [14132]
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 19. Summary. Page 29
Problem number : Ex 5
Date solved : Thursday, October 02, 2025 at 09:15:39 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 62

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

\begin{align*} y &= \frac {1}{\left (3 \,{\mathrm e}^{x}+c_1 \right )^{{1}/{3}} x} \\ y &= -\frac {1+i \sqrt {3}}{2 \left (3 \,{\mathrm e}^{x}+c_1 \right )^{{1}/{3}} x} \\ y &= \frac {i \sqrt {3}-1}{2 \left (3 \,{\mathrm e}^{x}+c_1 \right )^{{1}/{3}} x} \\ \end{align*}

✓ Mathematica. Time used: 10.83 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to \frac {1}{\sqrt [3]{x^3 \left (3 e^x+c_1\right )}}\\ y(x)&\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{x^3 \left (3 e^x+c_1\right )}}\\ y(x)&\to \frac {(-1)^{2/3}}{\sqrt [3]{x^3 \left (3 e^x+c_1\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.766 (sec). Leaf size: 73

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

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