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Question Number 212612 by hardmath last updated on 18/Oct/24

$$\mathrm{Solve}\mathrm{the}\mathrm{differential}\mathrm{equation}:$$$$\frac{\mathrm{dy}}{\mathrm{dx}}=\cos (\mathrm{x}+\mathrm{y})$$

Answered by mr W last updated on 19/Oct/24

$$lett=x+y$$$$\frac{dt}{dx}=1+\frac{dy}{dx}$$$$\frac{dt}{dx}-1=\cos t$$$$\frac{dt}{\cos t+1}=dx$$$$\int \frac{dt}{\cos t+1}=\int dx$$$$\int \frac{dt}{2{\cos }^2\frac{t}{2}}=\int dx$$$$\tan \frac{t}{2}=x+C$$$$t=2{\tan }^{-1}(x+C)$$$$x+y=2{\tan }^{-1}(x+C)$$$$\Rightarrow y=2{\tan }^{-1}(x+C)-x✓$$

Commented by hardmath last updated on 19/Oct/24

$$\mathrm{thankyou}\mathrm{dearprofessor}$$

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