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Question Number 190851 by Rupesh123 last updated on 13/Apr/23

Answered by mr W last updated on 13/Apr/23

$$\cos y^{\prime }=\sin y$$$$\sin (\frac{\pi }{2}-y^{\prime })=\sin y$$$$\frac{\pi }{2}-y^{\prime }=n\pi +{(-1)}^{n}y$$$$-y^{\prime }={(-1)}^{n}y+\frac{(2n-1)\pi }{2}$$$$-\frac{dy}{dx}={(-1)}^{n}y+\frac{(2n-1)\pi }{2}$$$$\frac{dy}{{(-1)}^{n}y+\frac{(2n-1)\pi }{2}}=-dx$$$$\int \frac{dy}{{(-1)}^{n}y+\frac{(2n-1)\pi }{2}}=-\int dx$$$$\ln [{(-1)}^{n}y+\frac{(2n-1)\pi }{2}]=-{(-1)}^{n}x+C_1$$$${(-1)}^{n}y+\frac{(2n-1)\pi }{2}={Ce}^{{(-1)}^{n+1}x}$$$$\Rightarrow y={(-1)}^n[{Ce}^{{(-1)}^{n+1}x}-\frac{(2n-1)\pi }{2}],n\in \mathbb{Z}$$

Commented by Rupesh123 last updated on 13/Apr/23

Excellent, sir!

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