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Using Pythagoras, x² + y² = r².Using basic trigonometry, x = rsinθ and y = rcosθ.
xy + 1 = r²sinθcosθ + 1 = (1/2)r²sin2θ + 1
Subbing in bot...

note: y' is often written as dy/dx, this doesn't change the question at all so no need to worry. Same for y'' and d²y/dx².All second-order differential equations that don't have 0 on...

This is the type of question which requires some out of the box thinking. A sketch is presented below: We start by defining A = (a_ij), B = (b_ij) and C = (c_i...

Consider the expression (cos(theta) + i*sin(theta))⁵ . (Where theta a real parameter).By De Moirve's theorem, we know this expression is equivalent to cos(5theta) + i sin(5theta).We can also ap...

The system is described by a homogeneous, second order differential equation d²x/dt² +(dx/dt * c/m) + kx/m =0. First, substitute the known constants (m,k,c) to get d²x/dt<...