Which of the following equation(s) is/are lin | vedclass

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(D) differential equation is linear if the dependent variable and its derivatives appear only in the first degree and are not multiplied together.$1$.

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$(A), (B), 	ext{ and } (C)$ all

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(D) differential equation is linear if the dependent variable and its derivatives appear only in the first degree and are not multiplied together.$1$. For option $(A)$: $\frac{dy}{dx} + \frac{y}{x} = \ln x$. This is a first-order linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = \frac{1}{x}$ and $Q = \ln x$. Thus,it is linear.$2$. For option $(B)$: $\frac{d^2y}{dx^2} = \cos x$. This is a second-order linear differential equation because the dependent variable $y$ and its derivative $\frac{d^2y}{dx^2}$ are of the first degree. Thus,it is linear.$3$. For option $(C)$: $dx + dy = 0$. This can be rewritten as $1 + \frac{dy}{dx} = 0$ or $\frac{dy}{dx} = -1$. This is a first-order linear differential equation. Thus,it is linear.Since all three equations satisfy the definition of a linear differential equation,the correct answer is $(D)$.

Which of the following equation$(s)$ is/are linear?

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