Internal problem ID [486]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.2. Page 48
Problem number: 8.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime }-\frac {x^{2}}{1+y^{2}}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 268
dsolve(diff(y(x),x) = x^2/(1+y(x)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {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 )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\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 )^{\frac {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 )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (4 x^{3}+12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}+4}\right )^{\frac {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 )^{\frac {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 )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.246 (sec). Leaf size: 307
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 {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{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 ) \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 {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}}-\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*}