Internal problem ID [3593]
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]
\[ \boxed {x {y^{\prime }}^{2}+a +b x -y-y b=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 59
dsolve(x*diff(y(x),x)^2+a+b*x-y(x)-b*y(x) = 0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left ({\left (\operatorname {RootOf}\left (\textit {\_Z} -\textit {\_Z}^{\frac {1}{b}} \left (\frac {c_{1}}{x}\right )^{\frac {-1+b}{2 b}}+1-b \right )+1\right )}^{2}+b \right ) x}{-b -1}-\frac {a}{-b -1} \]
✓ Solution by Mathematica
Time used: 90.22 (sec). Leaf size: 1197
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 {2 (b+1) \left (-\log \left (\sqrt {-a+b y(x)-b x+y(x)}+\sqrt {x}\right )+b \log \left (\sqrt {-a+b y(x)-b x+y(x)}+b \sqrt {x}\right )+b \log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )-\log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )+\log \left (\sqrt {x} \sqrt {-a+b y(x)+y(x)}-\sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+b x-y(x)\right )-b \log \left (b \left (\sqrt {x} \left (\sqrt {-a+b y(x)+y(x)}-\sqrt {-a+b y(x)-b x+y(x)}\right )-y(x)+x\right )+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-y(x)\right )\right )}{b^2-1}-\frac {2 (b+1) \left ((b-1) \log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )+\log \left (\sqrt {x} \sqrt {-a+b y(x)+y(x)}-\sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+b x-y(x)\right )-b \log \left (b \left (\sqrt {x} \left (\sqrt {-a+b y(x)+y(x)}-\sqrt {-a+b y(x)-b x+y(x)}\right )-y(x)+x\right )+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-y(x)\right )\right )}{b^2-1}=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 (b+1) \left (-\log \left (\sqrt {-a+b y(x)-b x+y(x)}-\sqrt {x}\right )+b \log \left (\sqrt {-a+b y(x)-b x+y(x)}-b \sqrt {x}\right )+b \log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )-\log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )+\log \left (-\sqrt {x} \sqrt {-a+b y(x)+y(x)}+\sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+b x-y(x)\right )-b \log \left (b \left (\sqrt {x} \left (\sqrt {-a+b y(x)-b x+y(x)}-\sqrt {-a+b y(x)+y(x)}\right )-y(x)+x\right )+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-y(x)\right )\right )}{b^2-1}-\frac {2 (b+1) \left ((b-1) \log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )+\log \left (-\sqrt {x} \sqrt {-a+b y(x)+y(x)}+\sqrt {x} \sqrt {-a+b y(x)-b x+y(x)}+\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-b y(x)+b x-y(x)\right )-b \log \left (b \left (\sqrt {x} \left (\sqrt {-a+b y(x)+y(x)}-\sqrt {-a+b y(x)-b x+y(x)}\right )+y(x)-x\right )-\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}-a+y(x)\right )\right )}{b^2-1}=c_1,y(x)\right ] \\ \end{align*}