Question

Solve each of the following problems by direct integration
a) 4(x) = 3t x(0) =2

Answer 1

To solve the problem 4(x) = 3t with x(0) = 2 by direct integration, we integrate both sides of the equation, apply the initial condition, and rearrange to find the function x(t) that satisfies the equation. The final solution is x = (3/8)t^2 - 2.

Step-by-step explanation:

To solve the problem 4(x) = 3t with x(0) = 2 by direct integration, we need to find the function x(t) that satisfies the initial condition. We can do this by integrating both sides of the equation.

∫4(x) dx = ∫3t dt

Applying the integral, we get 4x + C1 = (3/2)t^2 + C2, where C1 and C2 are constants of integration.

Now, we can substitute the initial condition x(0) = 2 to find the values of the constants. 4(2) + C1 = (3/2)(0)^2 + C2, which simplifies to 8 + C1 = C2.

Substituting this back into the equation, we have 4x + 8 = (3/2)t^2 + C1. Rearranging the equation for x, we get x = (3/8)t^2 - 2.