29.30.6 problem 864
Internal
problem
ID
[5446]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
30
Problem
number
:
864
Date
solved
:
Monday, January 27, 2025 at 11:24:10 AM
CAS
classification
:
[[_homogeneous, `class C`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}+a +b x -y-b y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.089 (sec). Leaf size: 116
dsolve(x*diff(y(x),x)^2+a+b*x-y(x)-b*y(x) = 0,y(x), singsol=all)
\[
y \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z} \sqrt {\frac {x}{c_{1}}}-b \sqrt {\frac {x}{c_{1}}}-\textit {\_Z}^{\frac {1}{b}} \left (\frac {x}{c_{1}}\right )^{\frac {1}{2 b}}+\sqrt {\frac {x}{c_{1}}}\right )^{2} x +2 \operatorname {RootOf}\left (\textit {\_Z} \sqrt {\frac {x}{c_{1}}}-b \sqrt {\frac {x}{c_{1}}}-\textit {\_Z}^{\frac {1}{b}} \left (\frac {x}{c_{1}}\right )^{\frac {1}{2 b}}+\sqrt {\frac {x}{c_{1}}}\right ) x +\left (b +1\right ) x +a}{b +1}
\]
✓ Solution by Mathematica
Time used: 147.912 (sec). Leaf size: 1133
DSolve[x (D[y[x],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 (b-1) \text {arctanh}\left (\frac {\sqrt {-a+b y(x)+y(x)}}{\sqrt {-a+b y(x)-b x+y(x)}}\right )+2 \text {arctanh}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{\sqrt {x}}\right )-2 b \text {arctanh}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{b \sqrt {x}}\right )-b \log (-a-(b+1) (b x-y(x)))+\log ((b+1) (y(x)-x)-a)+2 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 )-2 \log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) y(x)\right )+2 (b-1) \int \frac {\sqrt {-a+b y(x)-b x+y(x)}}{\left (\sqrt {-a+b y(x)-b x+y(x)}+\sqrt {x}\right ) \left (\sqrt {-a+b y(x)-b x+y(x)}+b \sqrt {x}\right )} \, d\sqrt {-a+b y(x)-b x+y(x)}\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 {(b+1) \left (2 (b-1) \text {arctanh}\left (\frac {\sqrt {-a+b y(x)+y(x)}}{\sqrt {-a+b y(x)-b x+y(x)}}\right )-2 \text {arctanh}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{\sqrt {x}}\right )+2 b \text {arctanh}\left (\frac {\sqrt {-a+b y(x)-b x+y(x)}}{b \sqrt {x}}\right )-b \log (a+(b+1) (b x-y(x)))+\log ((b+1) (y(x)-x)-a)-2 \log \left (\sqrt {x}-\sqrt {-a+b y(x)-b x+y(x)}\right )+2 b \log \left (b \sqrt {x}-\sqrt {-a+b y(x)-b x+y(x)}\right )+2 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 )-2 \log \left (\sqrt {-a+b y(x)+y(x)} \sqrt {-a+b y(x)-b x+y(x)}+a-(b+1) 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*}