Internal problem ID [7811]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 231.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {\left (a y+x b +c \right ) y^{\prime }+\alpha y+\beta x +\gamma =0} \end {gather*}
✓ Solution by Maple
Time used: 0.672 (sec). Leaf size: 207
dsolve((a*y(x)+b*x+c)*diff(y(x),x)+alpha*y(x)+beta*x+gamma=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {-b \gamma +\beta c +\frac {\left (x \left (a \beta -b \alpha \right )+a \gamma -\alpha c \right ) \left (\sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \tan \left (\RootOf \left (\sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \ln \left (\frac {\left (x \left (a \beta -b \alpha \right )+a \gamma -\alpha c \right )^{2} \left (4 a \beta \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-\alpha ^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-2 \alpha b \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-b^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}\right )}{4 a}\right )+2 c_{1} \sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}+2 \textit {\_Z} \alpha -2 \textit {\_Z} b \right )\right )+\alpha +b \right )}{2 a}}{-a \beta +b \alpha } \]
✓ Solution by Mathematica
Time used: 1.612 (sec). Leaf size: 260
DSolve[(a*y[x]+b*x+c)*y'[x]+\[Alpha]*y[x]+\[Beta]*x+\[Gamma]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {(b-\alpha )^2 \left (-\frac {2 \text {ArcTan}\left (\frac {\frac {2 (a (\gamma +\beta x)-\alpha b x+\alpha (-c))}{a y(x)+b x+c}+\alpha -b}{(\alpha -b) \sqrt {\frac {4 (a \beta -\alpha b)}{(b-\alpha )^2}-1}}\right )}{\sqrt {\frac {4 (a \beta -\alpha b)}{(b-\alpha )^2}-1}}-\log \left (\frac {(a y(x)+b x+c) \left ((a (\gamma +\beta x)-\alpha b x+\alpha (-c)) \left (\frac {a (\gamma +\beta x)-\alpha b x+\alpha (-c)}{a y(x)+b x+c}+\alpha -b\right )-(\alpha b-a \beta ) (a y(x)+b x+c)\right )}{(-a (\gamma +\beta x)+\alpha b x+\alpha c)^2}\right )\right )}{2 (a \beta -\alpha b)}=\frac {(b-\alpha )^2 \log (a (\gamma +\beta x)-\alpha b x+\alpha (-c))}{a \beta -\alpha b}+c_1,y(x)\right ] \]