75.3.12 problem 12

Internal problem ID [19916]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 12
Date solved : Thursday, October 02, 2025 at 05:01:09 PM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 3 y^{2} y^{\prime }+y^{3}&=x -1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 61

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

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

✓ Mathematica. Time used: 5.462 (sec). Leaf size: 71

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

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

✓ Sympy. Time used: 1.161 (sec). Leaf size: 63

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

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