82.11.5 problem Ex. 5

Internal problem ID [18694]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Exercises at page 28
Problem number : Ex. 5
Date solved : Monday, March 31, 2025 at 05:58:06 PM
CAS classification : [_rational, _Bernoulli]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 82

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

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

✓ Mathematica. Time used: 4.205 (sec). Leaf size: 92

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

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

✓ Sympy. Time used: 4.388 (sec). Leaf size: 119

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

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