** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

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Apply the property of the product of two powers of the same base in reverse: $a^{m+n}=a^m\cdot a^n$

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Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

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Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

We can solve the integral $\int\frac{1}{e^{2y}}dy$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $2y$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Now, in order to rewrite $dy$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

Isolate $dy$ in the previous equation

Substituting $u$ and $dy$ in the integral and simplify

Apply the formula: $\int\frac{1}{e^x}dx$$=-\frac{1}{e^x}+C$, where $x=u$

Simplify $-\frac{1}{2}\left(\frac{1}{e^u}\right)$

Replace $u$ with the value that we assigned to it in the beginning: $2y$

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Solve the integral $\int\frac{1}{e^{2y}}dy$ and replace the result in the differential equation

We can solve the integral $\int e^{3x}dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it $u$), which when substituted makes the integral easier. We see that $3x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

Isolate $dx$ in the previous equation

Substituting $u$ and $dx$ in the integral and simplify

Take the constant $\frac{1}{3}$ out of the integral

Divide $1$ by $3$

The integral of the exponential function is given by the following formula $\displaystyle \int a^xdx=\frac{a^x}{\ln(a)}$, where $a > 0$ and $a \neq 1$

Replace $u$ with the value that we assigned to it in the beginning: $3x$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

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Solve the integral $\int e^{3x}dx$ and replace the result in the differential equation

Take the reciprocal of both sides of the equation

Move up the $-1$ from the denominator

Divide both sides of the equation by $-2$

Simplify $-2\left(\frac{1}{3}e^{3x}+C_0\right)$ using algebra

Multiply $-2$ times $\frac{1}{3}$

We can take out the unknown from the exponent by applying natural logarithm to both sides of the equation

Apply the formula: $\ln\left(e^x\right)$$=x$, where $x=2y$

Simplify the logarithm $\ln\left(\frac{1}{-\frac{2}{3}e^{3x}+C_1}\right)$

Divide both sides of the equation by $2$

Simplifying the quotients

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Find the explicit solution to the differential equation. We need to isolate the variable $y$

** Final Answer

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