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

Question

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.

Answer 1

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

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

Answer 2

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