48.1.2 problem Example 3.2

Internal problem ID [7503]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.2 FIRST ORDER ODE. Page 114
Problem number : Example 3.2
Date solved : Monday, January 27, 2025 at 03:02:48 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 264

dsolve(diff(y(x),x)=x^2/(1-y(x)^2),y(x), singsol=all)
 

\begin{align*} y &= \frac {\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}}+4}{2 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{1}/{3}}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}}-4 i \sqrt {3}+4}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{1}/{3}}} \\ y &= \frac {i \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}} \sqrt {3}-4 i \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}}-4}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.581 (sec). Leaf size: 320

DSolve[D[y[x],x]==x^2/(1-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}-\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right )}{2^{2/3} \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ \end{align*}