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: 0.125 (sec). Leaf size: 271
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 \ln \left (x \right )-\frac {x \left (\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}\right )^{\frac {3}{2}}}{2 y \left (x \right )}-\operatorname {arctanh}\left (\frac {y \left (x \right )+x}{x \sqrt {\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )+\sqrt {\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}}-\frac {3 \sqrt {\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}}\, y \left (x \right )}{2 x}-\frac {x}{2 y \left (x \right )}-c_{1} = 0 \ln \left (x \right )+\frac {x \left (\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}\right )^{\frac {3}{2}}}{2 y \left (x \right )}+\operatorname {arctanh}\left (\frac {y \left (x \right )+x}{x \sqrt {\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \left (x \right )}{x}\right )-\sqrt {\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}}+\frac {3 \sqrt {\frac {x^{2}+2 y \left (x \right ) x -3 y \left (x \right )^{2}}{x^{2}}}\, y \left (x \right )}{2 x}-\frac {x}{2 y \left (x \right )}-c_{1} = 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*}