Final answer:
To solve the given initial value problem (IVP), we can separate the variables and integrate both sides. The explicit solution to the IVP is x(t) = -4^(-5/5) + 1/2 - (1/2)cos(t^2).
Step-by-step explanation:
To solve the given initial value problem (IVP), we can separate the variables and integrate both sides. Let's start by dividing both sides of the equation by (2x^6) and multiplying by dt to isolate dx. This gives us:
dx/(2x^6) = -tsin(t^2)dt
Next, we integrate both sides with respect to their respective variables:
∫dx/(2x^6) = ∫-tsin(t^2)dt
Integrating the left side gives us: x^-5/5 = -1/2cos(t^2) + C
Applying the initial condition x(0) = -4, we can find the value of the constant C. Substituting x = -4 and t = 0 into the equation gives us:
(-4)^-5/5 = -1/2cos(0) + C
Simplifying, we find that C = -4^(-5/5) + 1/2.
Therefore, the explicit solution to the IVP is:
x(t) = -4^(-5/5) + 1/2 - (1/2)cos(t^2)