To solve the differential equation dy/dx = 2y e^(3x), we can use the method of separation of variables. This technique allows us to rearrange the equation to isolate the variables on either side.
First, we start with the original equation:
dy/dx = 2y e^(3x)
Next, we can rewrite it to separate the variables:
dy/y = 2e^(3x) dx
Now, we can integrate both sides. The left side will give us:
∫(1/y) dy = ln|y| + C₁
And for the right side, we integrate:
∫(2e^(3x)) dx = (2/3)e^(3x) + C₂
Now, we combine both results:
ln|y| = (2/3)e^(3x) + C
Where C = C₂ – C₁ is the constant of integration. To solve for y, we exponentiate both sides:
|y| = e^((2/3)e^(3x) + C) = e^C e^((2/3)e^(3x)
We can represent e^C as another constant K (since it is still a constant):
y = K e^((2/3)e^(3x)
Finally, we arrive at the general solution of the differential equation:
y = K e^((2/3)e^(3x)
In this solution, K can take any real value, allowing us to cover all possible solutions of the differential equation.