Question

Find an implicit and an explicit solution of the given initial-value problem. (Use x for x(t).) dx dt = 3(x2 + 1), x(π/4) = 1

Answer 1

To find an implicit and explicit solution to the initial-value problem dx/dt = 3(x^2 + 1), x(π/4) = 1, integrate both sides of the equation, substitute the initial condition to find the constant, and then rewrite the equation to obtain implicit and explicit solutions.

Step-by-step explanation:

To find an implicit and explicit solution to the initial-value problem dx/dt = 3(x^2 + 1), x(π/4) = 1, we can solve it as follows:

1. Integrate both sides of the equation with respect to t: ∫ dx/dt dt = ∫ 3(x^2 + 1) dt

   This gives us x = x^3 + 3t + C.

2. Substitute the initial condition x(π/4) = 1 to find the value of the constant C:

   1 = (1^3 + 3(π/4) + C)

   Solving for C, we get C = 1 - (π/4) - 1 = -π/4.

3. Substitute the value of C back into the equation:

   x = x^3 + 3t - π/4.

Therefore, the implicit solution is x = x^3 + 3t - π/4 and the explicit solution is x = (3t - π/4 - 1)^(1/3).