Internal problem ID [4965]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page
7
Problem number: 5.
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: 353
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: 2.064 (sec). Leaf size: 234
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\&,2\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*}