Internal problem ID [2889]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 5
Problem number: 141.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
Solve \begin {gather*} \boxed {3 y^{\prime }-x -\sqrt {x^{2}-3 y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 234
dsolve(3*diff(y(x),x) = x+sqrt(x^2-3*y(x)),y(x), singsol=all)
\[ \frac {2 x^{2} \sqrt {x^{2}-3 y \relax (x )}}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x )^{2} \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}-\frac {2 x^{3}}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x )^{2} \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}-\frac {6 \sqrt {x^{2}-3 y \relax (x )}}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x ) \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}+\frac {9 x}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x ) \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.166 (sec). Leaf size: 498
DSolve[3 y'[x]==x+Sqrt[x^2-3 y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{12} \left (x^2+\frac {x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}+\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (2 x^2+\frac {\left (-1-i \sqrt {3}\right ) x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{24} \left (2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+216 e^{3 c_1}\right )}{\sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{x^6-540 e^{3 c_1} x^3+24 \sqrt {3} \sqrt {e^{3 c_1} \left (-x^3+27 e^{3 c_1}\right ){}^3}-5832 e^{6 c_1}}\right ) \\ \end{align*}