Solved example of evaluate logarithms
Apply the property of the product of two powers of the same base in reverse: $a^{m+n}=a^m\cdot a^n$
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
Integrate both sides, 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 part of the integral with a new variable, which when substituted makes the integral easier. We see that $e^{2y}$ 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
Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function
Substitute $u$ back with the value that we assigned to it: $e^{2y}$
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 part of the integral with a new variable, 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
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 $a \neq 1$
Substitute $u$ back with the value that we assigned to it: $3x$
Solve the integral $\int e^{3x}dx$ and replace the result in the differential equation
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Eliminate the $-\frac{1}{2}$ from the left, multiplying both sides of the equation by $$
Solve the product $-2\left(\frac{1}{3}e^{3x}+C_0\right)$
We can rename $-2C_0$ as other constant
We can take out the unknown from the exponent by applying natural logarithm to both sides of the equation
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
Calculating the natural logarithm of $e$
Any expression multiplied by $1$ is equal to itself
Divide both sides of the equation by $-2$
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