What is the solution for derivative of z(x*e^{2x})=e^{2x} ?

The solution for derivative of z(x*e^{2x})=e^{2x} is z(x)= 1/2+(c_{1})/(e^{2x)}

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derivative of z(x*e^{2x})=e^{2x}

\frac{d}{d x} (z (x) \cdot e^{2 x}) = e^{2 x}

z (x) = \frac{1}{2} + \frac{c _{1}}{e ^{2 x}}

Solution steps

\frac{d}{d x} (z (x) \cdot e^{2 x}) = e^{2 x}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

z (x) e^{2 x} = \int e^{2 x} d x

\int e^{2 x} d x = \frac{1}{2} e^{2 x} + c_{1}

z (x) e^{2 x} = \frac{1}{2} e^{2 x} + c_{1}

Isolate

z (x) : z (x) = \frac{1}{2} + \frac{c _{1}}{e ^{2 x}}

z (x) = \frac{1}{2} + \frac{c _{1}}{e ^{2 x}}

e^{y} \cdot d y - 2 x \cdot d x = 0

\frac{d y}{d t} + 24 \cdot 1 0^{- 3} y = 384 \cdot 1 0^{- 3}

\frac{d y}{d x} (\frac{x ^{2}}{3 y ^{2}}) = 1

f (x) = \frac{1}{x ^{7}} \frac{df}{d x}

x \cdot y^{^{'}} = 3 y

What is the solution for derivative of z(x*e^{2x})=e^{2x} ?
The solution for derivative of z(x*e^{2x})=e^{2x} is z(x)= 1/2+(c_{1})/(e^{2x)}