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(D) The given differential equation is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = a$ and $Q = e^{mx}$.The Integrat

Vedclass · Accepted Answer

$(a + m)y = e^{mx} + c e^{-ax}(a + m)$

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(D) The given differential equation is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = a$ and $Q = e^{mx}$.The Integrating Factor $(I.F.)$ is given by $I.F. = e^{\int P dx} = e^{\int a dx} = e^{ax}$.The general solution is $y \cdot (I.F.) = \int Q \cdot (I.F.) dx + C$.Substituting the values,we get $y \cdot e^{ax} = \int e^{mx} \cdot e^{ax} dx + C$.$y \cdot e^{ax} = \int e^{(a+m)x} dx + C$.$y \cdot e^{ax} = \frac{e^{(a+m)x}}{a+m} + C$.Dividing both sides by $e^{ax}$,we get $y = \frac{e^{mx}}{a+m} + C e^{-ax}$.Multiplying by $(a+m)$,we get $(a+m)y = e^{mx} + C(a+m)e^{-ax}$.Thus,the correct option is $(d)$.

The solution of the differential equation $\frac{dy}{dx} + ay = e^{mx}$ is

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