29.21.28 problem 604

Internal problem ID [5198]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 21
Problem number : 604
Date solved : Monday, January 27, 2025 at 10:20:36 AM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+2 y x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 325

dsolve((a^2+x^2+y(x)^2)*diff(y(x),x)+2*x*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -\frac {2 \left (a^{2}+x^{2}-\frac {\left (-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+4 x^{6}+9 c_{1}^{2}}\right )^{{2}/{3}}}{4}\right )}{\left (-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+4 x^{6}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{4}+\frac {1}{4}\right ) \left (-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+4 x^{6}+9 c_{1}^{2}}\right )^{{2}/{3}}+\left (a^{2}+x^{2}\right ) \left (i \sqrt {3}-1\right )}{\left (-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+4 x^{6}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+4 x^{6}+9 c_{1}^{2}}\right )^{{2}/{3}}}{4}+\left (a^{2}+x^{2}\right ) \left (1+i \sqrt {3}\right )}{\left (-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+4 x^{6}+9 c_{1}^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.316 (sec). Leaf size: 317

DSolve[(a^2+x^2+y[x]^2)D[y[x],x]+2 x y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {4 \left (a^2+x^2\right )^3+9 c_1{}^2}+3 c_1\right ){}^{2/3}-2 a^2-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+9 c_1{}^2}+3 c_1}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (a^2+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+9 c_1{}^2}+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a^2+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^2+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+9 c_1{}^2}+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\ y(x)\to 0 \\ \end{align*}