29.5.24 problem 141

Internal problem ID [4741]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 5
Problem number : 141
Date solved : Monday, January 27, 2025 at 09:36:03 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 3 y^{\prime }&=x +\sqrt {x^{2}-3 y} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 108

dsolve(3*diff(y(x),x) = x+sqrt(x^2-3*y(x)),y(x), singsol=all)
 

\[ \frac {2 \left (x^{2}-3 y \left (x \right )\right )^{{3}/{2}} \left (c_{1} y \left (x \right )^{2} x^{2}-4 c_{1} y \left (x \right )^{3}+1\right )+2 x \left (x^{2}-\frac {9 y \left (x \right )}{2}\right ) \left (c_{1} y \left (x \right )^{2} x^{2}-4 c_{1} y \left (x \right )^{3}-1\right )}{\left (x^{2}-4 y \left (x \right )\right ) y \left (x \right )^{2} \left (-2 \sqrt {x^{2}-3 y \left (x \right )}+x \right ) \left (x +\sqrt {x^{2}-3 y \left (x \right )}\right )^{2}} = 0 \]

✓ Solution by Mathematica

Time used: 60.184 (sec). Leaf size: 499

DSolve[3 D[y[x],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 {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}}}+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*}