47.1.5 problem 5

Internal problem ID [7386]
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
Date solved : Monday, January 27, 2025 at 02:51:34 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: 2.294 (sec). Leaf size: 307

DSolve[D[y[x],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*}