Internal problem ID [4493]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson
10
Problem number: Recognizable Exact Differential equations. Integrating factors. Exercise
10.18, page 90.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\[ \boxed {2 x y+\left (x^{2}+y^{2}+a \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 313
dsolve((2*x*y(x))+(x^2+y(x)^2+a)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x^{2}-4 a}{2 \left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+\left (x^{2}+a \right ) \left (i \sqrt {3}-1\right )}{\left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (x^{2}+a \right ) \left (1+i \sqrt {3}\right )}{\left (-12 c_{1} +4 \sqrt {4 x^{6}+12 a \,x^{4}+12 x^{2} a^{2}+4 a^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.319 (sec). Leaf size: 299
DSolve[(2*x*y[x])+(x^2+y[x]^2+a)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}-2 a-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (a+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a+x^2\right )^3+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}