Find the particular solution of dy/dx = 4y, given y(0) = 2.

Find the particular solution of dy/dx = 4y, given y(0) = 2.

Step 1: Start with the differential equation dy/dx = 4y.
Step 2: Recognize that this is a separable differential equation.
Step 3: Rewrite the equation as dy/y = 4 dx.
Step 4: Integrate both sides: ∫(1/y) dy = ∫4 dx.
Step 5: The left side integrates to ln|y| and the right side integrates to 4x + C, where C is a constant.
Step 6: So, we have ln|y| = 4x + C.
Step 7: Exponentiate both sides to solve for y: y = e^(4x + C).
Step 8: Rewrite e^(4x + C) as y = e^(4x) * e^C. Let C' = e^C, so y = C'e^(4x).
Step 9: This gives us the general solution: y = Ce^(4x), where C is a constant.
Step 10: Use the initial condition y(0) = 2 to find C. Substitute x = 0 into the general solution: y(0) = Ce^(4*0) = C.
Step 11: Set C equal to 2 because y(0) = 2. So, C = 2.
Step 12: Substitute C back into the general solution: y = 2e^(4x).

Separation of Variables – The method used to solve the differential equation by separating the variables y and x.
Initial Conditions – Applying the initial condition to find the specific constant in the general solution.
Exponential Functions – Understanding the behavior of exponential functions in the context of differential equations.