Internal problem ID [3347]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 21
Problem number: 605.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+b^{2}+x^{2}+2 y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 900
dsolve((a^2+x^2+y(x)^2)*diff(y(x),x)+b^2+x^2+2*x*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 \left (a^{2}+x^{2}\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {a^{2}+x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 a^{2}+2 x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {a^{2}+x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 a^{2}+2 x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 c_{1}+4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 x^{3} c_{1}+9 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.253 (sec). Leaf size: 438
DSolve[(a^2+x^2+y[x]^2)y'[x]+b^2+x^2+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+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+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+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+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+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+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+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}{2 \sqrt [3]{2}} \\ \end{align*}