40.5.5 problem 21

Internal problem ID [6670]
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
Date solved : Monday, January 27, 2025 at 02:18:47 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} 8 y {y^{\prime }}^{2}-2 x y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.031 (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 &= -\frac {\sqrt {2}\, x}{4} \\ y &= \frac {\sqrt {2}\, x}{4} \\ y &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {8 y^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_1 &= 0 \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {8 y^{2}-x^{2}}{x^{2}}}}\right )+\ln \left (\frac {y}{x}\right )-c_1 &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.370 (sec). Leaf size: 174

DSolve[8*y[x]*D[y[x],x]^2-2*x*D[y[x],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*}