Internal problem ID [11636]
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]
\[ \boxed {2 y^{2}+\left (4 y x -y^{2}\right ) y^{\prime }=-x^{2}} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 439
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 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {8 x^{2} c_{1}^{2}}{\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x}{c_{1}} \\ y \left (x \right ) &= \frac {-\frac {\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {4 x^{2} c_{1}^{2}}{\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x -\frac {i \sqrt {3}\, \left (-16 c_{1}^{2} x^{2}+\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {2}{3}}\right )}{4 \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}}{c_{1}} \\ y \left (x \right ) &= -\frac {16 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {2}{3}}+16 c_{1}^{2} x^{2}-8 c_{1} x \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}+\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {2}{3}}}{4 \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 33.481 (sec). Leaf size: 731
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 {\left (1-i \sqrt {3}\right ) \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 \sqrt [3]{2}}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) 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 {\left (1+i \sqrt {3}\right ) \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 \sqrt [3]{2}}-\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) 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 {8 \sqrt [3]{2} x^2+4 \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3} x+2^{2/3} \left (\sqrt {33} \sqrt {x^6}+17 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}} \\ y(x)\to \frac {8 i \sqrt [3]{2} \sqrt {3} x^2-8 \sqrt [3]{2} x^2+8 \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3} x-i 2^{2/3} \sqrt {3} \left (\sqrt {33} \sqrt {x^6}+17 x^3\right )^{2/3}-2^{2/3} \left (\sqrt {33} \sqrt {x^6}+17 x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}} \\ y(x)\to \frac {\left (\sqrt {33} \sqrt {x^6}+17 x^3\right )^{2/3} \text {Root}\left [2 \text {$\#$1}^3-1\&,3\right ]-4 \sqrt [3]{-2} x^2+2 \sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3} x}{\sqrt [3]{\sqrt {33} \sqrt {x^6}+17 x^3}} \\ \end{align*}