Internal problem ID [10618]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number: 23(a).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {x^{2}+2 y^{2}+\left (4 x y-y^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 441
dsolve((x^2+2*y(x)^2)+(4*x*y(x)-y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\frac {\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {8 c_{1}^{2} x^{2}}{\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 x c_{1}}{c_{1}} \\ y \left (x \right ) = \frac {-\frac {\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {4 c_{1}^{2} x^{2}}{\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 x c_{1} -\frac {i \sqrt {3}\, \left (\frac {\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {8 c_{1}^{2} x^{2}}{\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \left (x \right ) = \frac {-\frac {\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {4 c_{1}^{2} x^{2}}{\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 x c_{1} +\frac {i \sqrt {3}\, \left (\frac {\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}-\frac {8 c_{1}^{2} x^{2}}{\left (4+68 x^{3} c_{1}^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 31.117 (sec). Leaf size: 564
DSolve[(x^2+2*y[x]^2)+(4*x*y[x]-y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{17 x^3+\sqrt {33 x^6+34 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^2}{\sqrt [3]{17 x^3+\sqrt {33 x^6+34 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+2 x \\ y(x)\to \frac {1}{2} \left ((-2)^{2/3} \sqrt [3]{17 x^3+\sqrt {33 x^6+34 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}-\frac {8 \sqrt [3]{-2} x^2}{\sqrt [3]{17 x^3+\sqrt {33 x^6+34 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x\right ) \\ y(x)\to x \left (2+\frac {4 (-1)^{2/3} \sqrt [3]{2} x}{\sqrt [3]{17 x^3+\sqrt {33 x^6+34 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}\right )-\sqrt [3]{-\frac {1}{2}} \sqrt [3]{17 x^3+\sqrt {33 x^6+34 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}} \\ y(x)\to \frac {\sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}}+2 x \\ y(x)\to -\sqrt [3]{-\frac {1}{2}} \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}+\frac {4 (-1)^{2/3} \sqrt [3]{2} x^2}{\sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}}+2 x \\ y(x)\to \frac {1}{2} \left ((-2)^{2/3} \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}-\frac {8 \sqrt [3]{-2} x^2}{\sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}}+4 x\right ) \\ \end{align*}