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Find the general solution of \frac{dy}{dx}+ay =e^{mx}

Answers (1)

   Given:

\frac{dy}{dx}+ay=e^{mx}\\

It is a first order differential equation. Comparing it with,

      \frac{dy}{dx}+p(x)y=q(x)\\

      P(x) =a

      Q(x)=exm

      Calculating Integrating Factor

  IF=e^{\int p(x)dx}\\ IF=e^{\int a \;dx}\\ IF=e^{ax}

      Hence the solution of the given differential equation is ,

      y\left (IF \right )=\int q(x).(IF)dx\\ y.(e^{ax})=\int e^{mx}e^{ax} dx\\ y.(e^{ax})=\int e^{\left (m+a \right )x} dx\\ Formula: \int e^{ax}dx=\frac{1}{a}e^{ax}\\ y.(e^{ax})=\frac{\left (e^{(m+a)x} \right )}{m+a}+c

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