Internal problem ID [7659]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 78.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]
Solve \begin {gather*} \boxed {y^{\prime }+a \sin \left (\alpha y+\beta x \right )+b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.118 (sec). Leaf size: 118
dsolve(diff(y(x),x) + a*sin(alpha*y(x)+beta*x) + b=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {-\beta x +2 \arctan \left (\frac {\tan \left (\frac {c_{1} \sqrt {-a^{2} \alpha ^{2}+\alpha ^{2} b^{2}-2 \alpha b \beta +\beta ^{2}}}{2}-\frac {x \sqrt {-a^{2} \alpha ^{2}+\alpha ^{2} b^{2}-2 \alpha b \beta +\beta ^{2}}}{2}\right ) \sqrt {-a^{2} \alpha ^{2}+\alpha ^{2} b^{2}-2 \alpha b \beta +\beta ^{2}}-a \alpha }{b \alpha -\beta }\right )}{\alpha } \]
✓ Solution by Mathematica
Time used: 0.487 (sec). Leaf size: 86
DSolve[y'[x]+ a*Sin[\[Alpha]*y[x]+\[Beta]*x] + b==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\beta x+2 \operatorname {ArcTan}\left (\frac {-a \alpha +\sqrt {(\beta -\alpha b)^2-a^2 \alpha ^2} \tan \left (\frac {1}{2} (-x+c_1) \sqrt {(\beta -\alpha b)^2-a^2 \alpha ^2}\right )}{\alpha b-\beta }\right )}{\alpha } \\ \end{align*}