26.3.11 problem Exact Differential equations. Exercise 9.15, page 79

Internal problem ID [6944]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number : Exact Differential equations. Exercise 9.15, page 79
Date solved : Tuesday, September 30, 2025 at 04:07:07 PM
CAS classification : [_exact, _Bernoulli]

\begin{align*} {\mathrm e}^{x} \left (y^{3}+x y^{3}+1\right )+3 y^{2} \left (x \,{\mathrm e}^{x}-6\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.250 (sec). Leaf size: 37

ode:=exp(x)*(y(x)^3+x*y(x)^3+1)+3*y(x)^2*(x*exp(x)-6)*diff(y(x),x) = 0; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\left (-\left ({\mathrm e}^{x}+5\right ) \left (x \,{\mathrm e}^{x}-6\right )^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 x \,{\mathrm e}^{x}-12} \]

✓ Mathematica. Time used: 0.843 (sec). Leaf size: 28

ode=Exp[x]*(y[x]^3+x*y[x]^3+1)+3*y[x]^2*(x*Exp[x]-6)*D[y[x],x]==0; 
ic=y[0]==1; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\sqrt [3]{-e^x-5}}{\sqrt [3]{e^x x-6}} \end{align*}

✓ Sympy. Time used: 1.743 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x*exp(x) - 18)*y(x)**2*Derivative(y(x), x) + (x*y(x)**3 + y(x)**3 + 1)*exp(x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt [3]{\frac {- e^{x} - 5}{x e^{x} - 6}} \]