The height x metres, of a column of water in a fountain display satisfies the differential equation dx/dt = 8sin(2t)/(3sqrt(x)), where t is the time in seconds after the display begins. (a) Solve the differential equation, given that x(0)=0

Our DE (Differential Equation) dx/dt = 8sin(2t)/(3sqrt(x)) is separable because dx/dt can be expressed as a product of two functions of the form dx/dt = f(t)g(x) = (8sin(2t)) * (1/(3sqrt(x))).To solve the DE, we first divide both sides of our equation by our function using the x variable: 3sqrt(x) * dx/dt = 8sin(2t), we can now integrate both sides with respect to x | 3sqrt(x) *dx/dt * dt = | 8sin(2t)dt, pull out the constants 3| sqrt(x)dx = 8| sin(2t)dt, integrate both sides, without forgetting the constant of integration C, 3 * ((2/3) * x^(3/2)) = 8 * ((1/2) * -cos(2t)) + C, simplify the expression2x^(3/2) = -4cos(2t) + C, x^(3/2) = -2cos(2t) + C/2, x = (-2cos(2t) + C/2)^(2/3). Now to find the value of C/2, we plug in our initial condition x(0)=0, 0 = (-2cos(0) +C/2)^(2/3), C/2 = 2. So finally, x = (2-2cos(2t))^(2/3).