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Answer: Thus  : $y=A\cos x+\sin x$   is the solution of the given differential equation

Hint: Solve first order derivative and put values in differential equation to be verified.

Given:$y=A\cos x+\sin x,\cos x\frac{dy}{dx}+\left ( sinx \right )y=1$

Solution:

$\frac{dy}{dx}=A\sin x+\cos x$ …Differentiating w.r.t to x,

Now, differential eq.

\begin{aligned} &\text { L. } H . S=\operatorname{cos} x\left(\frac{d y}{d x}\right)+(\operatorname{sin} x) y \\ &\operatorname{cos} x(-A \operatorname{sin} x+\operatorname{cos} x)+\operatorname{sin} x(A \operatorname{cos} x+\operatorname{sin} x) \\ &-A \operatorname{sin} x \operatorname{cos} x+\operatorname{cos}^{2} x+A \operatorname{sin} x \operatorname{cos} x+\operatorname{sin}^{2} x \\ &\operatorname{cos}^{2} x+\operatorname{sin}^{2} x=1=R . H . S \ldots\left\{\operatorname{cos}^{2} x+\operatorname{sin}^{2} x=1\right\} \end{aligned}

Hence, Proved.

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