# I get a different answer on this separable differential equation than I'm supposed to.

by user773674   Last Updated May 23, 2020 01:20 AM

I let $$y = v'$$, But when I try and do it i get $$v = e^x + e^{c_1} + c_2.$$ Can someone please show me? $$v'' -v' =0$$

=> $$v=C_1e^x +C_2$$, this is supposed to be it.

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$$v''-v'=0$$ $$y'-y=0$$ $$(ye^{-x})'=0$$ Integrate $$ye^{-x}=c_1$$ $$v'=c_1e^{x}$$ Integrate again: $$v=c_1e^x+c_2$$

I used integrating factor for solving this DE: $$y'-y=0$$ But it's also separable: $$y'=y$$ $$\dfrac {y'}{y}=1$$ $$\int \dfrac {dy}{y}=\int dx$$ $$\ln y= x+c_1$$ $$y=e^{x+c_1}$$ $$y=ce^x$$

As $$e^{c_1} + c_2 = C(An \ arbitary \ constant)$$ The family of curves is $$y = e^x + C$$ which satisfies $$y^{\prime} = e^x$$