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: 0.007 (sec). Leaf size: 234
dsolve(3*diff(y(x),x) = x+sqrt(x^2-3*y(x)),y(x), singsol=all)
\[ \frac {2 x^{2} \sqrt {x^{2}-3 y \relax (x )}}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x )^{2} \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}-\frac {6 \sqrt {x^{2}-3 y \relax (x )}}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x ) \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}-\frac {2 x^{3}}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x )^{2} \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}+\frac {9 x}{\left (-x^{2}+4 y \relax (x )\right ) y \relax (x ) \left (2 \sqrt {x^{2}-3 y \relax (x )}-x \right ) \left (x +\sqrt {x^{2}-3 y \relax (x )}\right )^{2}}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.807 (sec). Leaf size: 552
DSolve[3 y'[x]==x+Sqrt[x^2-3 y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {Root}\left [432 \text {$\#$1}^6-216 \text {$\#$1}^5 x^2+27 \text {$\#$1}^4 x^4-216 \text {$\#$1}^3 e^{6 c_1}+378 \text {$\#$1}^2 e^{6 c_1} x^2-144 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+27 e^{12 c_1}\&,1\right ] \\ y(x)\to \text {Root}\left [432 \text {$\#$1}^6-216 \text {$\#$1}^5 x^2+27 \text {$\#$1}^4 x^4-216 \text {$\#$1}^3 e^{6 c_1}+378 \text {$\#$1}^2 e^{6 c_1} x^2-144 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+27 e^{12 c_1}\&,2\right ] \\ y(x)\to \text {Root}\left [432 \text {$\#$1}^6-216 \text {$\#$1}^5 x^2+27 \text {$\#$1}^4 x^4-216 \text {$\#$1}^3 e^{6 c_1}+378 \text {$\#$1}^2 e^{6 c_1} x^2-144 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+27 e^{12 c_1}\&,3\right ] \\ y(x)\to \text {Root}\left [432 \text {$\#$1}^6-216 \text {$\#$1}^5 x^2+27 \text {$\#$1}^4 x^4-216 \text {$\#$1}^3 e^{6 c_1}+378 \text {$\#$1}^2 e^{6 c_1} x^2-144 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+27 e^{12 c_1}\&,4\right ] \\ y(x)\to \text {Root}\left [432 \text {$\#$1}^6-216 \text {$\#$1}^5 x^2+27 \text {$\#$1}^4 x^4-216 \text {$\#$1}^3 e^{6 c_1}+378 \text {$\#$1}^2 e^{6 c_1} x^2-144 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+27 e^{12 c_1}\&,5\right ] \\ y(x)\to \text {Root}\left [432 \text {$\#$1}^6-216 \text {$\#$1}^5 x^2+27 \text {$\#$1}^4 x^4-216 \text {$\#$1}^3 e^{6 c_1}+378 \text {$\#$1}^2 e^{6 c_1} x^2-144 \text {$\#$1} e^{6 c_1} x^4+16 e^{6 c_1} x^6+27 e^{12 c_1}\&,6\right ] \\ y(x)\to 0 \\ \end{align*}