Marlosampagaidn
Answered

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Which of the following is the particular solution of:
dydx=2y=6

Select one:
a. y=3−2e−2x

b. y=−3+2e−2x

c. y=−3−2e−2x

d. y=3+2e−2x

Sagot :

Yankeriiiidn

The particular solution of dy/dx = 2y + 6 is:

b. y = -3 + 2e^(-2x)

Explanation:

- The given differential equation is dy/dx = 2y + 6.

- To solve this linear first-order differential equation, we can apply the method of integrating factors.

- The integrating factor for this equation is e^(∫2dx) = e^(2x).

- Multiplying the given differential equation by the integrating factor, we get e^(2x)dy/dx - 2ye^(2x) = 6e^(2x).

- This can be rewritten as d(ye^(2x))/dx = 6e^(2x).

- Integrating both sides with respect to x gives ye^(2x) = ∫6e^(2x)dx = 3e^(2x) + C, where C is the constant of integration.

- Solving for y, we get y = 3 + Ce^(-2x).

- To find the particular solution, we use the initial condition that when x = 0, y = -3.

- Plugging in these values, we find that the constant C = -3.

- Thus, the particular solution is y = -3 + 2e^(-2x).

Therefore, the correct answer is option b: y = -3 + 2e^(-2x).