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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...
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Answered by James M. • Maths tutor
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Derive double angle formulas from addition formulae

sin2x = 2sinxcosxcos2x = cos 2 x - sin 2 x=2 cos 2 x - 1= 1 - 2sin 2 xtan2x = (2tanx)/(1-tan 2 x)
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Answered by Alexander H. • Maths tutor
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Find the equation of the normal to the curve x^3 + 2(x^2)y = y^3 + 15 at the point (2, 1)

The first thing when seeing such a question is to find out what the question wants you to do and plan your method to do so.In this case, the question asks you to find the normal of a curve at a given point (...
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Answered by Kelvin L. • Maths tutor
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The equation kx^2+4kx+5=0, where a is a constant, has no real roots. Find the range of possible values of k.

First, if k=0 then the equation has no real roots since in this case we would have 5=0 (this is a proof by contradiction). A quadratic equation of the form ax^2+bx+c=0 has no real roots if the discriminant b...
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Answered by Emily N. • Maths tutor
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How do you differentiate?

Say you have the term y=3X 2 . You start by bringing the power down and placing it in front of the 3 so they multiply like so y=2x3X 2 then next you take one away from the power like so y=2x3X 1 . This can t...
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Answered by Alex Y. • Maths tutor
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