ODE No. 78

3.78 ODE No. 78

$\frac{d}{d x} y (x) + a \sin (α y (x) + β x) + b = 0$

Mathematica: cpu = 0.860609 (sec), leaf count = 1317 ${{y (x) \to \frac{2 \tan^{- 1} (\frac{\frac{a^{2} \sqrt{- (a α + b α - β) (a α - b α + β)} \tan (\frac{1}{2} (\frac{a^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{b^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{a^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{b^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{2 b β x α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{2 b β c_{1} α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{β^{2} x}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{β^{2} c_{1}}{\sqrt{- (a α + b α - β) (a α - b α + β)}})) α^{2}}{(a α + b α - β) (a α - b α + β)} - \frac{b^{2} \sqrt{- (a α + b α - β) (a α - b α + β)} \tan (\frac{1}{2} (\frac{a^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{b^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{a^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{b^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{2 b β x α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{2 b β c_{1} α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{β^{2} x}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{β^{2} c_{1}}{\sqrt{- (a α + b α - β) (a α - b α + β)}})) α^{2}}{(a α + b α - β) (a α - b α + β)} - a α + \frac{2 b β \sqrt{- (a α + b α - β) (a α - b α + β)} \tan (\frac{1}{2} (\frac{a^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{b^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{a^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{b^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{2 b β x α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{2 b β c_{1} α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{β^{2} x}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{β^{2} c_{1}}{\sqrt{- (a α + b α - β) (a α - b α + β)}})) α}{(a α + b α - β) (a α - b α + β)} - \frac{β^{2} \sqrt{- (a α + b α - β) (a α - b α + β)} \tan (\frac{1}{2} (\frac{a^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{b^{2} x α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{a^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{b^{2} c_{1} α^{2}}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{2 b β x α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{2 b β c_{1} α}{\sqrt{- (a α + b α - β) (a α - b α + β)}} - \frac{β^{2} x}{\sqrt{- (a α + b α - β) (a α - b α + β)}} + \frac{β^{2} c_{1}}{\sqrt{- (a α + b α - β) (a α - b α + β)}}))}{(a α + b α - β) (a α - b α + β)}}{α b - β}) - β x}{α}}}$

Maple: cpu = 0.655 (sec), leaf count = 118 ${y (x) = \frac{1}{α} (- β x + 2 \arctan (\frac{\tan (1 / 2_C 1 \sqrt{- a^{2} α^{2} + α^{2} b^{2} - 2 α b β + β^{2}} - 1 / 2 x \sqrt{- a^{2} α^{2} + α^{2} b^{2} - 2 α b β + β^{2}}) \sqrt{- a^{2} α^{2} + α^{2} b^{2} - 2 α b β + β^{2}} - a α}{b α - β}))}$

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