How can I find the general solution of the differential equation dy/dx = 2y e^(3x)?

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To find the general solution of the differential equation dy/dx = 2y e^(3x), we can begin by recognizing that this is a first-order linear ordinary differential equation.

The equation can be rearranged to separate the variables:

dy/y = 2 e^(3x) dx

Now, we can integrate both sides. The left side, dy/y, integrates to ln|y|, and the right side integrates as follows:

∫2 e^(3x) dx = (2/3)e^(3x) + C

Let’s put this all together:

ln|y| = (2/3)e^(3x) + C

To eliminate the natural logarithm, we exponentiate both sides:

|y| = e^((2/3)e^(3x) + C)

We can express the constant term e^C as another constant K (where K = e^C, K > 0). This gives us:

y = Ke^((2/3)e^(3x})

Since K can be positive or negative, we can drop the absolute value and simply write:

y = Ke^((2/3)e^(3x}

Thus, the general solution to the differential equation dy/dx = 2y e^(3x) is:

y = Ke^((2/3)e^(3x)

where K is any real constant.

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