Solved example of linear differential equation
Rewrite the differential equation using Leibniz notation
We can identify that the differential equation has the form: $\frac{dy}{dx} + P(x)\cdot y(x) = Q(x)$, so we can classify it as a linear first order differential equation, where $P(x)=2$ and $Q(x)=0$. In order to solve the differential equation, the first step is to find the integrating factor $\mu(x)$
Compute the integral
The integral of a constant is equal to the constant times the integral's variable
To find $\mu(x)$, we first need to calculate $\int P(x)dx$
So the integrating factor $\mu(x)$ is
Any expression multiplied by $0$ is equal to $0$
Now, multiply all the terms in the differential equation by the integrating factor $\mu(x)$ and check if we can simplify
We can recognize that the left side of the differential equation consists of the derivative of the product of $\mu(x)\cdot y(x)$
Integrate both sides of the differential equation with respect to $dx$
Simplify the left side of the differential equation
The integral of a constant is equal to the constant times the integral's variable
Solve the integral $\int0dx$ 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 $C$
$x+0=x$, where $x$ is any expression
Multiply the equation by the reciprocal of $e^{2x}$
Find the explicit solution to the differential equation
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