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).

JM
Answered by James M. Maths tutor

5928 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

How do I integrate tan^2 x?


find the integral of y=x^2 +sin^2(x) with respect to x between the limits 0 and pi


Find the inverse of the function g(x)=(4+3x)/(5-x)


C2 differentiate 2x^2 -3x +4 with respect to X


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences