Internal problem ID [5344]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page
74
Problem number: 16.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {2 y-{y^{\prime }}^{2}-4 x y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 700
dsolve(2*y(x)=diff(y(x),x)^2+4*x*diff(y(x),x),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-2 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right ) \left (\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+6 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right )}{8 \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right ) \left (i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-12 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right )}{32 \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x^{2}-12 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}\right )}{32 \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.241 (sec). Leaf size: 1344
DSolve[2*y[x]==y'[x]^2+4*x*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (-x^2+\frac {x \left (x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}+\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{4} \left (-2 x^2-\frac {\left (1+i \sqrt {3}\right ) x \left (x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{4} \left (-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6+5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (-16 \sqrt {2} x^9+24 e^{3 c_1} x^6-6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{2} \left (-x^2+\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}+\frac {x^4-2 \sqrt {2} e^{3 c_1} x}{\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}\right ) \\ y(x)\to \frac {1}{4} \left (-2 x^2+\frac {\left (1+i \sqrt {3}\right ) x \left (-x^3+2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\ y(x)\to \frac {1}{4} \left (-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x \left (x^3-2 \sqrt {2} e^{3 c_1}\right )}{\sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}}-\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^6-5 \sqrt {2} e^{3 c_1} x^3+\sqrt {e^{3 c_1} \left (16 \sqrt {2} x^9+24 e^{3 c_1} x^6+6 \sqrt {2} e^{6 c_1} x^3+e^{9 c_1}\right )}+e^{6 c_1}}\right ) \\ \end{align*}