Internal problem ID [5082]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x^{2}}{1-y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 357
dsolve(diff(y(x),x)=x^2/(1-y(x)^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-4 x^{3}-12 c_{1}+4 \sqrt {x^{6}+6 x^{3} c_{1}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.369 (sec). Leaf size: 246
DSolve[y'[x]==x^2/(1-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2+\sqrt [3]{2} \left (-x^3+\sqrt {-4+\left (x^3-3 c_1\right ){}^2}+3 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{-x^3+\sqrt {-4+\left (x^3-3 c_1\right ){}^2}+3 c_1}} \\ y(x)\to \frac {-2 \sqrt [3]{-2}+(-2)^{2/3} \left (-x^3+\sqrt {-4+\left (x^3-3 c_1\right ){}^2}+3 c_1\right ){}^{2/3}}{2 \sqrt [3]{-x^3+\sqrt {-4+\left (x^3-3 c_1\right ){}^2}+3 c_1}} \\ y(x)\to \frac {\text {Root}\left [\text {$\#$1}^3-2\&,3\right ]}{\sqrt [3]{-x^3+\sqrt {-4+\left (x^3-3 c_1\right ){}^2}+3 c_1}}-\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-x^3+\sqrt {-4+\left (x^3-3 c_1\right ){}^2}+3 c_1} \\ \end{align*}