Finding the General Solution of the Differential Equation
To find the general solution of the differential equation given by
dy/dx = y * e^(2x)
we can use the method of separation of variables. Here’s a step-by-step approach:
Step 1: Separate the Variables
We start by rewriting the equation to isolate y and x on opposite sides:
dy/y = e^(2x) dx
Step 2: Integrate Both Sides
Next, we integrate both sides. The left side is the integral of 1/y and the right side is the integral of e^(2x):
∫(1/y) dy = ∫e^(2x) dx
This results in:
ln|y| = (1/2)e^(2x) + C
where C is the constant of integration.
Step 3: Solve for y
To solve for y, we exponentiate both sides:
|y| = e^((1/2)e^(2x) + C)
This can be simplified to:
y = ±e^C * e^(1/2)e^(2x).
Step 4: Simplifying the Solution
Letting K = ±e^C, we can express the solution as:
y = K * e^(x^2)
Thus, the general solution of the differential equation dy/dx = y * e^(2x) is:
y = K * e^(x^2)
where K is an arbitrary constant.