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The Taylor expansion/theorem is used to express any function as a power series about a certain point. Using the given formula of the Taylor expansion, we can approximate a trigonometric function (for exam...

Note that |sin(x)| <= 1 for all real x, so our solution must be complex if one exists at all
Euler's formula gives that exp(ix) = cos(x) + isin(x), and hence exp(-ix) = cos(x) - isin(x) since cos ...

2x² < x+3, 2x²- x - 3 < 0, (2x - 3) (x + 1) < 0, Positive quadratic. Roots: x = -1 and x = 3/2, Therefore, x takes values greater than -1 and less tha...

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...