Internal problem ID [5327]
Book: Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres.
McGraw Hill 1952
Section: Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page
65
Problem number: 21.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {8 y {y^{\prime }}^{2}-2 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 95
dsolve(8*y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, x}{4} \\ y \left (x \right ) &= \frac {\sqrt {2}\, x}{4} \\ y \left (x \right ) &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {8 y \left (x \right )^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} &= 0 \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {8 y \left (x \right )^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.347 (sec). Leaf size: 174
DSolve[8*y[x]*y'[x]^2-2*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^{4 c_1} \sqrt {e^{8 c_1}-2 i x}}{2 \sqrt {2}} \\ y(x)\to \frac {e^{4 c_1} \sqrt {e^{8 c_1}-2 i x}}{2 \sqrt {2}} \\ y(x)\to -\frac {e^{4 c_1} \sqrt {2 i x+e^{8 c_1}}}{2 \sqrt {2}} \\ y(x)\to \frac {e^{4 c_1} \sqrt {2 i x+e^{8 c_1}}}{2 \sqrt {2}} \\ y(x)\to 0 \\ y(x)\to -\frac {x}{2 \sqrt {2}} \\ y(x)\to \frac {x}{2 \sqrt {2}} \\ \end{align*}