Question

Consider the initial value problem (IVP) given by: (10 - 6y + e^(-3x))dx - 2dy = 0, with initial condition y(0) = 7/6. What is the explicit solution of the IVP? a) y(x) = 1 - 2e^(-3x) + 3x b) y(x) = 1 + 2e^(-3x) + 3x c) y(x) = 1 - 2e^(3x) + 3x d) y(x) = 1 + 2e^(3x) + 3x

Answer 1

The explicit solution of the initial value problem (IVP) (10 - 6y + e^(-3x))dx - 2dy = 0, with initial condition y(0) = 7/6, is y(x) = 1 + 2e^(-3x) + 3x. Option B is answer.

Step-by-step explanation:

To solve the IVP, we can first rewrite the equation in the form:

dy/dx = (10 - 6y + e^(-3x)) / 2

This is a separable differential equation, which means we can separate the variables and integrate both sides:

∫ dy = ∫ (10 - 6y + e^(-3x)) / 2 dx

On the left side, we can use the substitution u = y^2 to integrate:

y^2 = 5x - 3y - e^(-3x) / 3 + C

where C is the constant of integration.

On the right side, we can use the substitution v = -3x to integrate:

y^2 = 5x - 3y + e^v / 3 + C

To solve for C, we can use the initial condition y(0) = 7/6:

(7/6)^2 = 5(0) - 3(7/6) + e^(0) / 3 + C

C = 9/2

Substituting this value of C back into the equation, we get the explicit solution:

y(x) = 1 + 2e^(-3x) + 3x

Option B is answer.