Consider the differential equation frac{dy}{d | vedclass

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(C) Given the differential equation: $\frac{dy}{dx} = \frac{1}{ax + 4y + 7}$.Taking the reciprocal, we get: $\frac{dx}{dy} = ax + 4y + 7$.Rearranging

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Only $B$ and $A$ are true

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(C) Given the differential equation: $\frac{dy}{dx} = \frac{1}{ax + 4y + 7}$.Taking the reciprocal, we get: $\frac{dx}{dy} = ax + 4y + 7$.Rearranging the terms, we have: $\frac{dx}{dy} - ax = 4y + 7$.This is a linear differential equation of the form $\frac{dx}{dy} + Px = Q$, where $P = -a$ and $Q = 4y + 7$.Since the equation can be written in the form $\frac{dx}{dy} + Px = Q$, it is linear in $x$.Because the equation is linear in $x$, it is not linear in $y$.Thus, statement $A$ is true and statement $B$ is true.The integrating factor $(IF)$ is given by $e^{\int P dy} = e^{\int -a dy} = e^{-ay}$.Therefore, statement $D$ is false, and statement $C$ is false.Hence, only $A$ and $B$ are true.

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