82.11.5 problem Ex. 5

Internal problem ID [18765]
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 : Tuesday, January 28, 2025 at 12:15:39 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*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 82

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

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

✓ Solution by Mathematica

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

DSolve[3*x*(1-x^2)*y[x]^2*D[y[x],x]+(2*x^2-1)*y[x]^3==a*x^3,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*}