Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.

Find the particular solution of dy/dx = 4y with the initial condition y(0) = 2.

Step 1: Start with the differential equation dy/dx = 4y.
Step 2: Recognize that this is a separable equation, meaning we can separate y and x.
Step 3: Rewrite the equation as dy/y = 4 dx.
Step 4: Integrate both sides. The left side becomes ln|y| and the right side becomes 4x + C, where C is a constant.
Step 5: After integration, we have ln|y| = 4x + C.
Step 6: Exponentiate both sides to solve for y. This gives us y = e^(4x + C).
Step 7: Rewrite e^(4x + C) as y = e^(4x) * e^C. Let C' = e^C, so y = C'e^(4x).
Step 8: The general solution is y = Ce^(4x), where C is a constant.
Step 9: Use the initial condition y(0) = 2 to find the value of C.
Step 10: 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.