60.1.230 problem 231
Internal
problem
ID
[10244]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
231
Date
solved
:
Monday, January 27, 2025 at 06:41:15 PM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} \left (a y+b x +c \right ) y^{\prime }+\alpha y+\beta x +\gamma &=0 \end{align*}
✓ Solution by Maple
Time used: 2.401 (sec). Leaf size: 236
dsolve((a*y(x)+b*x+c)*diff(y(x),x)+alpha*y(x)+beta*x+gamma=0,y(x), singsol=all)
\[
y = \frac {-\left (\left (\beta x +\gamma \right ) a +\left (-b x -c \right ) \alpha \right ) \sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \tan \left (\operatorname {RootOf}\left (-2 \sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \ln \left (2\right )+\sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}\right ) \left (a \beta x -\alpha b x +a \gamma -\alpha c \right )^{2}}{a}\right )+2 c_{1} \sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}+2 \textit {\_Z} \alpha -2 \textit {\_Z} b \right )\right )+\left (b x +c \right ) \alpha ^{2}+\left (\left (-\beta x -\gamma \right ) a +b \left (b x +c \right )\right ) \alpha -a \left (\left (\beta x -\gamma \right ) b +2 \beta c \right )}{2 a \left (a \beta -b \alpha \right )}
\]
✓ Solution by Mathematica
Time used: 1.777 (sec). Leaf size: 260
DSolve[(a*y[x]+b*x+c)*D[y[x],x]+\[Alpha]*y[x]+\[Beta]*x+\[Gamma]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {(b-\alpha )^2 \left (-\frac {2 \arctan \left (\frac {\frac {2 (a (\gamma +\beta x)-\alpha b x+\alpha (-c))}{a y(x)+b x+c}+\alpha -b}{(\alpha -b) \sqrt {\frac {4 (a \beta -\alpha b)}{(b-\alpha )^2}-1}}\right )}{\sqrt {\frac {4 (a \beta -\alpha b)}{(b-\alpha )^2}-1}}-\log \left (\frac {(a y(x)+b x+c) \left ((a (\gamma +\beta x)-\alpha b x+\alpha (-c)) \left (\frac {a (\gamma +\beta x)-\alpha b x+\alpha (-c)}{a y(x)+b x+c}+\alpha -b\right )-(\alpha b-a \beta ) (a y(x)+b x+c)\right )}{(-a (\gamma +\beta x)+\alpha b x+\alpha c)^2}\right )\right )}{2 (a \beta -\alpha b)}=\frac {(b-\alpha )^2 \log (a (\gamma +\beta x)-\alpha b x+\alpha (-c))}{a \beta -\alpha b}+c_1,y(x)\right ]
\]