The solution to the equation dy/dx + 8y = 4 when y(0) = 2 is y = 2 e8x.

## Explanation

The differential equation dy/dx + 8y = 4 is a first-order linear differential equation with constant coefficients. It can be solved using the integrating factor method. The integrating factor is e8x, and when this integrating factor is multiplied with the given equation, the equation becomes e8x(dy/dx + 8y) = 4e8x . Integrating both sides of the equation gives us e8xy = 4/8∫e8xdx + c, where c is the constant of integration.

## Solution

Since we are given the initial condition y(0) = 2, we can use this to calculate the value of c. Substituting x = 0 and y = 2 in e8xy = 4/8∫e8xdx + c, we get 2 = c. Thus, the solution is y = 2 e8x.

## Examples

• If x = 3, then y = 2 e24
• If x = -2, then y = 2 e-16
• If x = 0.5, then y = 2 e4

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