Internal problem ID [3594]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 864.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class C], _rational, _dAlembert]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+a +b x -y-b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.117 (sec). Leaf size: 77
dsolve(x*diff(y(x),x)^2+a+b*x-y(x)-b*y(x) = 0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (\RootOf \left (\textit {\_Z} \sqrt {\frac {x}{c_{1}}}-b \sqrt {\frac {x}{c_{1}}}-\textit {\_Z}^{\frac {1}{b}} \left (\frac {x}{c_{1}}\right )^{\frac {1}{2 b}}+\sqrt {\frac {x}{c_{1}}}\right )+1\right )^{2}+b \right ) x}{-1-b}-\frac {a}{-1-b} \]
✓ Solution by Mathematica
Time used: 58.586 (sec). Leaf size: 1050
DSolve[x (y'[x])^2+(a+b x-y[x])-b y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {(b+1) \left (2 \sqrt {-b} \text {ArcTan}\left (\frac {(b+1) \left (\sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+b (x-y(x))\right )+a b}{\sqrt {b} (-a+b y(x)+y(x))}\right )-\sqrt {b} \left (2 \sqrt {-b^2} \text {ArcTan}\left (\frac {(b+1) \left (\sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+b x-y(x)\right )+a}{\sqrt {b} (-a+b y(x)+y(x))}\right )+2 (b-1) \log \left ((b+1) \left (\sqrt {-a+b y(x)-b x+y(x)}-\sqrt {-b} \sqrt {x}\right )\right )+\log \left (b (a+(b+1) (x-y(x))) \left (2 \sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+2 b x-y(x)\right )\right )-b \log \left ((a+(b+1) (b x-y(x))) \left (2 \sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+2 b x-y(x)\right )\right )\right )\right )}{\sqrt {b} \left (b^2-1\right )}+\frac {(b+1) \left (\log (a+(b+1) (x-y(x)))-b \log (a+(b+1) (b x-y(x)))-2 \log \left (\sqrt {-a+b y(x)-b x+y(x)}+\sqrt {x}\right )+2 b \log \left (\sqrt {-a+b y(x)-b x+y(x)}+b \sqrt {x}\right )+2 \tanh ^{-1}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{\sqrt {x}}\right )-2 b \tanh ^{-1}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{b \sqrt {x}}\right )\right )}{b^2-1}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {(b+1) \left (\sqrt {-b} \left (2 \sqrt {-b^2} \text {ArcTan}\left (\frac {(b+1) \left (\sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+b x-y(x)\right )+a}{\sqrt {b} (-a+b y(x)+y(x))}\right )-2 (b-1) \log \left ((b+1) \left (\sqrt {-a+b y(x)-b x+y(x)}-\sqrt {-b} \sqrt {x}\right )\right )-\log \left (b (a+(b+1) (x-y(x))) \left (2 \sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+2 b x-y(x)\right )\right )+b \log \left ((a+(b+1) (b x-y(x))) \left (2 \sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+2 b x-y(x)\right )\right )\right )+2 \sqrt {b} \text {ArcTan}\left (\frac {(b+1) \left (\sqrt {-b} \sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+b (x-y(x))\right )+a b}{\sqrt {b} (-a+b y(x)+y(x))}\right )\right )}{\sqrt {-b} \left (b^2-1\right )}+\frac {(b+1) \left (\frac {2 \sqrt {b} \sqrt {-b^2} \tanh ^{-1}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{b \sqrt {x}}\right )}{\sqrt {-b}}+\log (a+(b+1) (x-y(x)))-b \log (a+(b+1) (b x-y(x)))-2 \log \left (\sqrt {-a+b y(x)-b x+y(x)}-\sqrt {x}\right )+2 b \log \left (\sqrt {-a+b y(x)-b x+y(x)}-b \sqrt {x}\right )-2 \tanh ^{-1}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{\sqrt {x}}\right )\right )}{b^2-1}=c_1,y(x)\right ] \\ \end{align*}