5.9 problem 124

Internal problem ID [2873]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 5
Problem number: 124.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class C], _dAlembert]

Solve \begin {gather*} \boxed {y^{\prime }-a -b \sin \left (A x +B y\right )=0} \end {gather*}

Solution by Maple

Time used: 0.029 (sec). Leaf size: 115

dsolve(diff(y(x),x) = a+b*sin(A*x+B*y(x)),y(x), singsol=all)
 

\[ y \relax (x ) = -\frac {A x +2 \arctan \left (\frac {b B +\tan \left (\frac {c_{1} \sqrt {B^{2} a^{2}-b^{2} B^{2}+2 A B a +A^{2}}}{2}-\frac {x \sqrt {B^{2} a^{2}-b^{2} B^{2}+2 A B a +A^{2}}}{2}\right ) \sqrt {B^{2} a^{2}-b^{2} B^{2}+2 A B a +A^{2}}}{a B +A}\right )}{B} \]

Solution by Mathematica

Time used: 0.43 (sec). Leaf size: 84

DSolve[y'[x]==a+b Sin[A x+ B y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {-A x+2 \text {ArcTan}\left (\frac {-b B+\sqrt {(a B+A-b B) (B (a+b)+A)} \tan \left (\frac {1}{2} (x-c_1) \sqrt {(a B+A-b B) (B (a+b)+A)}\right )}{a B+A}\right )}{B} \\ \end{align*}