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(A) Given the equation: $c_{1} y = (c_{2} + c_{3}) e^{x + c_{4}}$We can simplify this by grouping the constants: $y = \left( \frac{c_{2} + c_{3}}{c_{1

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(A) Given the equation: $c_{1} y = (c_{2} + c_{3}) e^{x + c_{4}}$We can simplify this by grouping the constants: $y = \left( \frac{c_{2} + c_{3}}{c_{1}} \right) e^{c_{4}} \cdot e^{x}$Let $C = \left( \frac{c_{2} + c_{3}}{c_{1}} \right) e^{c_{4}}$,where $C$ is a single arbitrary constant.Thus,the equation becomes $y = C e^{x}$.Since there is only one independent arbitrary constant $C$,the order of the differential equation obtained by eliminating it is $1$.Differentiating $y = C e^{x}$ with respect to $x$,we get $\frac{dy}{dx} = C e^{x}$.Substituting $y = C e^{x}$ into the derivative,we get $\frac{dy}{dx} = y$.This is a first-order differential equation,so the order is $1$.

The order of the differential equation obtained by eliminating arbitrary constants in the family of curves $c_{1} y = (c_{2} + c_{3}) e^{x + c_{4}}$ is

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