Solving a First Order ODE | y’ = 2e^x using Separation of Variables

y’=2e^x

The given equation is y’ = 2e^x

The given equation is y’ = 2e^x. This is a first order ordinary differential equation, where y’ represents the derivative of y with respect to x.

To solve this equation, we can use the method of separation of variables. The idea is to separate the variables x and y on different sides of the equation and integrate both sides.

Starting with the equation y’ = 2e^x, we can rewrite it as dy/dx = 2e^x. Now, we separate the variables by moving all terms involving y to one side and all terms involving x to the other side:

1/y dy = 2e^x dx

Next, we integrate both sides. The integral of 1/y dy is ln|y|, and the integral of 2e^x dx is 2e^x + C, where C is the constant of integration. Thus, we have:

ln|y| = 2e^x + C

To eliminate the absolute value, we can rewrite the equation as:

y = ±e^(2e^x + C) = ±e^(2e^x)e^C = Ke^(2e^x)

Here, K is another constant that combines the previous constant of integration, C, and the positive/negative sign in front of e^(2e^x).

So, the general solution to the differential equation y’ = 2e^x is y = Ke^(2e^x), where K is any constant.

To determine the particular solution, we need an initial condition or boundary condition. This means we need a specific value of x and y to substitute into the general solution and find the corresponding value of K. This will give us the specific solution to the differential equation.

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