Internal problem ID [3676]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 32
Problem number: 950.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {y \left (y^{\prime }\right )^{2}-\left (x +y\right ) y^{\prime }+y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.161 (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 \relax (x ) = x \\ y \relax (x ) = 0 \\ \ln \relax (x )-\frac {x \left (\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}\right )^{\frac {3}{2}}}{2 y \relax (x )}-\arctanh \left (\frac {x +y \relax (x )}{x \sqrt {\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \relax (x )}{x}\right )+\sqrt {\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}}-\frac {3 y \relax (x ) \sqrt {\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}}}{2 x}-\frac {x}{2 y \relax (x )}-c_{1} = 0 \\ \ln \relax (x )+\frac {x \left (\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}\right )^{\frac {3}{2}}}{2 y \relax (x )}+\arctanh \left (\frac {x +y \relax (x )}{x \sqrt {\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}}}\right )+\ln \left (\frac {y \relax (x )}{x}\right )-\sqrt {\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}}+\frac {3 y \relax (x ) \sqrt {\frac {x^{2}+2 x y \relax (x )-3 y \relax (x )^{2}}{x^{2}}}}{2 x}-\frac {x}{2 y \relax (x )}-c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.115 (sec). Leaf size: 310
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 (-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}+\frac {4 y(x) \tanh ^{-1}\left (\frac {\sqrt {\frac {3 y(x)}{x}+1}}{\sqrt {\frac {3 y(x)}{x}-3}}\right )}{x}-1\right )}{4 y(x)}=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\ \text {Solve}\left [\tanh ^{-1}\left (\frac {\sqrt {\frac {3 y(x)}{x}+1}}{\sqrt {\frac {3 y(x)}{x}-3}}\right )+\frac {x \left (-i \sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {3 y(x)}{x}+1}+\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*}