If y=a^3 e^{b^2 x+c} is the general solution | vedclass

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(A) Given the general solution $y=a^3 e^{b^2 x+c}$.We can rewrite this as $y = (a^3 e^c) e^{b^2 x}$.Let $K = a^3 e^c$,where $K$ is an arbitrary consta

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$1$

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(A) Given the general solution $y=a^3 e^{b^2 x+c}$.We can rewrite this as $y = (a^3 e^c) e^{b^2 x}$.Let $K = a^3 e^c$,where $K$ is an arbitrary constant because $a$ and $c$ are arbitrary constants.Thus,the equation becomes $y = K e^{b^2 x}$.Now,differentiate with respect to $x$:$\frac{dy}{dx} = K e^{b^2 x} \cdot b^2$.Since $y = K e^{b^2 x}$,we substitute this back into the derivative:$\frac{dy}{dx} = b^2 y$.This is a first-order differential equation because it involves only the first derivative $\frac{dy}{dx}$.Therefore,the order of the differential equation is $1$.

If $y=a^3 e^{b^2 x+c}$ is the general solution of a differential equation,where $a$ and $c$ are arbitrary constants and $b$ is a fixed constant,then the order of the differential equation is

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