Internal problem ID [6886]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th
edition. 1997.
Section: CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number: 25.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 176
dsolve(y(x)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x \\ y \left (x \right ) &= 0 \\ \frac {-x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}+2 y \left (x \right ) \ln \left (\frac {y \left (x \right )}{x}\right )+\left (-2 \,\operatorname {arctanh}\left (\frac {x +y \left (x \right )}{x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}}\right )-2 c_{1} +2 \ln \left (x \right )\right ) y \left (x \right )-x}{2 y \left (x \right )} &= 0 \\ \frac {x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}+2 y \left (x \right ) \ln \left (\frac {y \left (x \right )}{x}\right )+\left (2 \,\operatorname {arctanh}\left (\frac {x +y \left (x \right )}{x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}}\right )-2 c_{1} +2 \ln \left (x \right )\right ) y \left (x \right )-x}{2 y \left (x \right )} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.268 (sec). Leaf size: 320
DSolve[y[x]*y'[x]^2-(x+y[x])*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {x \left (-i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}+\frac {4 y(x) \log \left (\sqrt {\frac {3 y(x)}{x}-3}-\sqrt {\frac {3 y(x)}{x}+1}\right )}{x}-\frac {4 y(x) \log \left (-i \left (\frac {3 y(x)}{x}+1\right )+i \sqrt {\frac {3 y(x)}{x}-3} \sqrt {\frac {3 y(x)}{x}+1}+\sqrt {2+2 i \sqrt {3}}\right )}{x}+1\right )}{4 y(x)}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [-\frac {x \left (i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}+\frac {4 y(x) \log \left (\sqrt {\frac {3 y(x)}{x}-3}-\sqrt {\frac {3 y(x)}{x}+1}\right )}{x}-\frac {4 y(x) \log \left (i \left (\frac {3 y(x)}{x}+1\right )-i \sqrt {\frac {3 y(x)}{x}-3} \sqrt {\frac {3 y(x)}{x}+1}+\sqrt {2-2 i \sqrt {3}}\right )}{x}+1\right )}{4 y(x)}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ y(x)\to 0 \\ \end{align*}